Wikipedia :
Group Theory
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Properties of Groups
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Simple groups
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Lorentz Group
Steiner systems
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Topological groups
Finite Simple Groups Classification (42:37)
Peter M. Neumann
(LMS, 1992).
The Symmetries of Things (1:12:13)
by John Conway (2012-08).
Finite Groups, Yesterday and Today (54:55)
Jean-Pierre Serre (2015年11月02日).
30 years of the Atlas and... before (46:40)
John G. Thompson (2015年11月02日).
Abstract group theory (21:57)
Grant Sanderson (3Blue1Brown, 2020年08月19日).
Bourbaki calls magma a set endowed with some internal well-defined operation. (Avoid the term groupoid for this, which is now best reserved for a concept in category theory.)
Multiplicative notations are often used where the binary operator is understood between consecutive symbols representing elements. Saying that an operation is multiplicative merely stresses the use of that convention usually supplemented by the optional use of a so-called multiplicative, symbol, (like a dot, an x-shaped cross, a delta or any special ad hoc symbol) whenever a clear separation between elements is deemed typographically appropriate.
Once a particular operator is so singled out, it's called a multiplication and the qualifier multiplicative can then be used, especially to distinguish that from co-existing additive concepts.
Occasionally, the multiplicative vocabulary is applied to several co-existing operators, making distinct multiplicative symbols mandatory. (Usage may or may not allow one such symbol to be dropped.) For example, the dot-product and the cross-product over 3D-vectors are both construed as multiplications but the cross symbol can never be dropped. In some international texts, that cross is replaced by a wedge, which is the general symbol for an exterior product, of which the ordinary cross-product can be construed to be a special case.
When that particular operation is not denoted by a cross symbol, it's called a vectorial product (Bourbaki: produit vectoriel, in French.)
If its operator is associative, a magma is called a semigroup. Associativity is the property which makes the use of parentheses optional:
x y z = (x y) z = x (y z)
The order of a finite semigroup is its number of elements. We count two semigroups as distinct when there's no isomorphism between them:
A semigroup operator may or may not be commutative. (In a commutative semigroup, xy is the same as yx for any pair of elements x and y.)
The numbers of distinct noncommutative semigroups are obtained from the above two tables, by termwise subtractions. Those numbers are always even because such semigroups come in pairs linked by an anti-isomorphism.
Lone operators are often designed to be associative. In complex structures with several operators, non-associativity may emerge in a natural way:
For example, in the realm of hypercomplex numbers, the multiplication of octonions or sedenions is not associative.
Likewise, the cross-product in ordinary three-dimensional vector space is not associative. Instead, it verifies what's called Jacobi's identity:
A × (B × C) = (A × B) × C + B × (A × C)
Incidentally, any vector space endowed with an anticommutative internal operator (×) verifying that identity is called a Lie algebra.
A semigroup in which there's a neutral element e is called a monoid :
$e , "x , e x = x e = x
The number of noncommutative monoids is obtained by subtracting the corresponding entries of the above tables. It's always an even number because such monoids come in pairs linked by an anti-isomorphism.
A monoid can also be defined as a category with a single object (the arrows of that category being the elements of the monoid).
Magma
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Quasigroup & Loop
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Semigroup &
Monoid
The semigroups of order 9 and their automorphism groups
by Andreas Distler & Tom Kelsey (2013年01月25日).
In a monoid, an element x is said to be right invertible if there's a right-inverse x' of x (which is to say that the product xx' is unity). It's called left invertible if there's a left-inverse x'' (such that x''x is unity).
When both inverses exist, they are necessarily equal (HINT: Consider x''xx' ). In this case, x is said to be invertible and its (unique) inverse is denoted x-1.
In a multiplicative monoid M, the set of all invertible elements form a group which is often denoted M*. (The neutral element of M belongs to M*.)
For example:
Cancellativity (cancellable element & cancellative operation)
All the finite strings (or words) whose characters (letters or symbols) are taken from a given alphabet form a monoid under the operation of concatenation (concatenating two strings means appending the second to [the right of] the first). The concatenation of two strings called A and B is best called "A before B".
The empty string is the neutral element for concatenation.
This monoid is free from any relations equating distinct strings of basic symbols. Hence the name (French: monoïde libre ).
Clearly, concatenating two nonempty strings yields something other than the empty string. The empty string is thus the only string with an inverse...
The free monoid over an alphabet of only one symbol is isomorphic to the natural integers endowed with addition (0,1,2,3...). In every other case, a free monoid is clearly not commutative.
Whenever some kind of associative multiplication is defined, something like x3 is simply shorthand for xxx. It's always legitimate to raise an element to the power of a positive integer that way.
The n-th power of an element can be well-defined even for non-associative multiplications. The weaker property of alternativity suffices (e.g., the multiplication of octonions is just alternative).
By definition, the weakest property of multiplication which allows a well-defined exponentiation (to the power of a positive integer) is called power associativity. Almost all "multiplications" have it.
x0 is always defined as equal to the neutral element, if there is one (otherwise, it's undefined). It makes no difference whether x is invertible or not (with ordinary arithmetic, zero to the power of zero equals one).
One enlightening example is that of the free monoids, defined in the previous section: xn denotes the concatenation of n strings identical to x. Thus, x0 denotes the empty string (the concatenation of no strings). x need not be "invertible".
If x is invertible, x-3 simply denotes (x-1)3. Only invertible elements can be raised to the power of a negative integer.
Raising something to the power of zero is a special case of an empty product. The result of not performing at all some well-defined associative operation depends on that operation alone: It's equal to its neutral element (whenever it has one).
Walther von Dyck (1856-1934) formally defined groups in 1882:
A group is a set G on which an internal operation is defined which verifies the following properties (using multiplicative notations for the operator).
G is called a commutative group (or abelian group) when we also have:
Niels Abel (1802-1829) considered algebraic equations whose Galois groups are commutative (and went on to prove that they are solvable). Leopold Kronecker (1823-1891) called such equations Abelian and the qualifier was first applied to all commutative groups by Camille Jordan (1838-1922) in 1870. The term is now so common in that capacity that it's usually lowercased (abelian).An additive group is merely a group (usually abelian) where additive notations are used: The plus sign (+) denotes the group operator.
Additive notations are rarely used for a noncommutative operator. (In a ring, addition is always commutative. In a near-ring, it doesn't have to be.) One well-known exception is the addition of transfinite ordinals à la Cantor (which I dare consider a misguided effort).
The double-sidedness of two of the above group axioms need not be postulated; it can be derived from one-sided equivalents of those axioms :
Indeed, we may compute x' x using just those two single-sided postulates:
x' x = x' x e = x' x x' (x' )' = x' e (x' )' = x' (x' )' = e
That would show x' to be the inverse of x, if we knew that e is neutral on both sides. That fact is easy to prove, using the above as a lemma:
"xÎG, e x = (x x' ) x = x (x' x) = x e = x
This double-sided neutrality implies that there's only one unity e . (HINT: Assume another unity e' and consider e e' ).
Similarly, there's only one inverse x' of x (HINT: Let x" be another and consider x' x x" ). So we may safely talk about the inverse of x.
Note, finally, that (x' )' = x (HINT: x' (x' )' = e ).
A subgroup H of a group G is a subset H of G which forms a group under the group operation defined over G. H is a subgroup of G if and only if it contains the product of any element of H by the inverse of any other element of H. A multiplicative subgroup is said to be stable by division.
"xÎH, "yÎH, x y-1 Î H
When additive nomenclature and notations are used, this translates into the following statement, which says that a subgroup of an additive group is merely a subset that's stable by subtraction :
"xÎH, "yÎH, x - y Î H
A proper subgroup of G is a subgroup of G not equal to G itself. The trivial group {e} has no proper subgroup.
Any intersection of subgroups is a subgroup.
The centralizer in a group G of a subset E consists of all the elements of G which commute with every element of E. It is a subgroup of G.
The centralizer in G of G itself is the center of G, denoted Z(G) (it's the intersection of all centralizers in G). The center is a normal subgroup of G, but other centralizers may not be. Elements of Z(G) are called central.
By definition:
For a left-ideal I,
the product ax is in I whenever x is:
"aÎA,
aI Í I
For a right-ideal I,
the product xa is in I whenever x is:
"aÎA,
Ia Í I
Unless otherwise specified,
an ideal is both a right-ideal and a left-ideal.
Note that the empty set is an ideal of any semigroup.
This is most often used in the context of ring theory, where an ideal of a ring (not necessarily a unital ring) is defined as being both a multiplicative ideal (in the above sense) and an additive subgroup (thus, the empty set is not an ideal of a ring because it's not a group).
Any intersection of [one-sided] ideals is a [one-sided] ideal. The intersection of all the ideals of a semigroup is called its minimal ideal. If it's nonempty, the minimal ideal M of a commutative semigroup is a group. This is to say that M has a neutral element, even if the whole semigroup doesn't.
For any subset E of a group G, the intersection of all subgroups of G containing E is a subgroup of G, called the subgroup generated by E.
E is said to be a set of generators of whatever subgroup it generates. A group which is generated by a finite set is said to be finitely generated.
For example, the additive group (Z,+) of the integers is generated by the set {1}. It's also generated by {2,3} or any other pair of coprime integers (because of Bezout's lemma). More generally, (Z,+) is generated by any set of coprime integers (not necessarily pairwise coprime) like {6,10,15}.
A finite group (of order n ) which is generated by a single element is a cyclic group. An element of such a group which generates the whole group is called a primitive element (or a primitive root, with the vocabulary inherited from representing the cyclic group of order n as the "n-th roots of unity" in complex numbers). There are f(n) different elements in a cyclic group which are primitive ones ( f being Euler's totient function).
The multiplicative group (Q+, ´) of positive rationals is not finitely generated. It's generated by the prime numbers {2,3,5,7,11,13,17,19...}.
Additive groups which are not finitely generated include the rationals, the reals, the complex numbers, the p-adic integers, the p-adic numbers, etc.
A finitely-generated group can be described by naming a set of generators and stating the nontrivial relations they obey (the relators). Those relators are normally given by expressions which are equal to the neutral element (minimally so) but explicit equations are also commonly used.
A free group has no relators. The simplest free group is isomorphic to the additive group of the integers (Z,+) and has the following multiplicative presentation, which names a single generator and states no relators:
< a |>
Less trivially, the octic group D4 could be presented as follows:
< r, s | r 4, s 2, srsr > or < r, s | r 4 = s 2 = srsr = 1 >
Do not confuse such presentations with (linear) representations.
By definition, the order |G| of a finite group G is its number of elements. (The order of an element x is the order of the subgroup generated by {x}.)
In a group G, the left-coset of an element x, with respect to the subgroup H, is the subset x H of G (consisting of all products x h where h is an element of H). Similarly, the right-coset is H x.
Two left-cosets with respect to H are either disjoint or identical and they have the same cardinality as H (i.e., the same number of elements if finite). Whenever it's finite, the number of left-cosets with respect to H is equal to the number of right-cosets. It's denoted [G:H] and is called the index of H in G.
In the case of a finite group G, the fact that such left-cosets form a partition of G shows that the order of the subgroup H divides evenly the order of G.
This result is known as Lagrange's Theorem. It's now presented as one of the most basic results of Group Theory, named in honor of Joseph-Louis Lagrange (1736-1813), who made a related remark in 1777. The general result was probably known to Cauchy (1789-1857) but it was only formally proved in 1861, by Camille Jordan (1838-1922; X1855).
Two subgroups are said to be commensurable when the index of their intersection is finite in each of them. The qualifier is inherited from ancient Greek mathematics, where two real numbers are called commensurable when they are proportional to two integers. The two additive groups generated by two such numbers are indeed commensurable in the above sense (their intersection is the additive group generated by the lowest common multiple of the two numbers).
Lagrange's theorem | Cosets and Lagrange's Theorem (9:18) by Liliana De Castro (Socratica, 2017年03月20日).
The commutative case can be used as a lemma to prove Cauchy's theorem and also its generalization by Sylow (Sylow's first theorem, 1872).
Lemma : Cauchy's group theorem holds for abelian groups.Proof : Let G be an abelian group G whose order is a multiple of the prime p: |G| = n = m×p. First, we see that the proposition is true if G is cyclic, generated by element a (since the element am is then of order p).
Otherwise, we proceed by induction on m, starting with the case m = 1 which makes p the order of G. This is trivial because, by Lagrange's theorem, the order of an element must divide the order of the group and can thus only be 1 or p. In other words, any element of G besides identity is a satisfactory element of order p (which establishes also that G is cyclic).
For m ≥ 2, consider an element h of G, besides identity. Let H be the nontrivial subgroup generated by h. H is a normal subgroup (as is any subgroup in the abelian case). Both H and G/H are nontrivial group of order strictly less than n (because we've already disposed of the cases where G is cyclic). Since the product of their orders (respectively |G| and [G:H]) equals n = m×p, at least one of them is divisible by p. In either case, the induction hypothesis implies that the corresponding group contains an element of order p. Either way, we can use that to obtain an element of order p in G, as follows:
This concludes the proof that Cauchy's theorem holds for abelian groups.
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Cauchy's group theorem (1845)
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Augustin-Louis Cauchy (1789-1857)
Cauchy's Theorem in Group Theory
by Robin Whitty
(Theorem of the Day #227).
For a prime p, a p-group is a group where the order of any element is a power of p. If it's a subgroup of G, it's called a p-subgroup of G.
In a finite group G, a Sylow p-subgroup (abbreviated p-SSG) is a maximal p-subgroup of G. The set of all p-SSG is denoted Sylp (G). Remarkably, all of those are isomorphic to each other.
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[画像: Come back later, we're still working on this one... ]
[画像: Come back later, we're still working on this one... ]
p-group
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Sylow theorems (1872)
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Peter Ludwig Sylow (1832-1918)
Groups of orders 56 are not simple (17:05)
by Suraj Pr. (Mr. SRJ, 2018年09月08日).
Visual Group Theory: The Sylow theorems (48:36)
by Matthew Macauley (2016年04月05日).
Sylow's theorems
by Robin Whitty
(Theorem of the Day #258).
The concept of a normal subgroup is due to Evariste Galois (1832).
A subgroup H is normal when aH = Ha for any a. Such a subgroup is also called invariant or distinguished (French: sous-groupe distingué ).
A subgroup H is normal iff it's stable under any inner isomorphism.
"aÎG, "xÎH, a x a-1 Î H
To a normal subgroup H corresponds an equivalence relation among elements of G defined by calling x and y equivalent when xy-1 is in H (in other words, when x and y have the same left cosets with respect to H).
The equivalence classes so defined form a group denoted G/H and called the quotient of H in G (or of G by H) also dubbed "G modulo H".
Although the above equivalence relation is defined for any subgroup H, the equivalence classes form a group only when H is normal.
Any group G is a normal subgroup of itself (the only non-proper one).
The trivial group {e} is a normal subgroup of any group G whose neutral element is e. (It's a proper subgroup of any such G but itself.)
The derived subgroup G' is also always a normal subgroup of G.
The center Z(G) of a group G consists of all the elements which commute with every element G. A member of Z(G) is called a central element. A noncentral element is an element which doesn't commute with at least one other element. The center is a normal subgroup. So is any subgroup of the center (in particular, any subgroup of an abelian group is normal).
If f is a homomorphism or an antihomomorphism from group G, then the kernel of f (ker f ) is a normal subgroup of G. More generally, so is the inverse image (pre-image) of any normal subgroup of f (G). For a normal subgroup H of G, the direct image f (H) is a normal subgroup of f (G).
For any subset E of the group G, the subgroup generated by all the conjugates of the elements of E is called conjugate closure of E. It's a normal subgroup containing E. In fact, it's the smallest normal subgroup containing E (i.e, it's the intersection of all normal subgroups containing E). It's thus also known as the normal closure of E.
The normalizer of a subgroup H consists of all elements x of the group G for which x H = H x (in particular all elements of H belong to its normalizer). The normalizer of H is a subgroup of G. By definition, H is a normal subgroup of its normalizer (H need not be a normal subgroup of the whole group G).
Two conventions are floating around to distinguish between a standard (reflexive) ordering relation and its strict (antireflexive) counterpart:
The above notations for normal subgroups were introduced by Helmut Wielandt around 1960. They are now also used to denote ideals in ring theory (since an ideal is to a ring what a normal subgroup is to a group).
Helmut Wielandt (1910-2001)
Notation
for proper normal subgroup
(or proper ideal)
TeX - LaTeX Stack Exchange
An homomorphism is a map (or function) which preserves some specific algebraic operation(s). A group homomorphism is thus a map f from a [multiplicative] group G into another group H, which is such that:
"xÎG, "yÎG, f (x y) = f (x) f (y)
If f is surjective ("onto" H) it's called an epihomomorphism (or "homomorphism onto"). If it's injective ("one-to-one") it's called an monomorphism. If it's bijective ("one-to-one onto") it's an isomorphism.
An homomorphism from G to itself is called an endomorphism of G. A bijective endomorphism is called an automorphism.
The automorphisms of a group G form a group, denoted Aut(G).
An anti-homomorphism, with respect to a multiplicative operator, is a function f which reverses the order of that multiplication :
"xÎG, "yÎG, f (x y) = f (y) f (x)
In any group, inversion is an example of an anti-homomorphism:
( x y ) -1 = y -1 x -1
The concepts defined above for homomorphisms have their counterparts for anti-homomorphisms: Anti-epihomomorphism, anti-monomomorphism, anti-isomorphism, anti-endomorphism and anti-automorphism.
For a homomorphism (or an anti-homomorphism) f from group G to a group of identity e, the kernel of f is a normal subgroup of G defined by
ker f = { xÎG | f (x) = e }
(The homomorphic pre-image of any normal subgroup is normal.)
Kernel of a ring homomorphism
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Categorial kernel
Kernel of a Group Homomorphism (4:52)
by Liliana De Castro (Socratica, 2016年05月02日).
A permutation of E is a one-to-one correspondence (bijection) of E onto itself. The term is most commonly used when E is finite, but it's also acceptable when E is infinite (possibly uncountably so).
The permutations of E are a group under function composition (o).
f o g (x) = f ( g (x) )
In the finite case, the symmetric group of degree n is denoted Sn. Its order is the number of permutations of n elements, namely n! ("n factorial").
The largest order of an element in the symmetric group Sn is traditionally denoted g(n) where g is called Landau's function, in honor of the German mathematician Edmund Landau (1877-1938) who proved the following asymptotic equivalence in 1902:
Log g(n) ~ (n Log n)½
Even permutations form the alternating group An (whose order is n!/2 ). It's the derived subgroup of the symmetric group: An = S'n
An even permutation is obtained by an even number of switches (swaps of two elements). The parity, or signature, of a finite permutation may be determined by counting its number of inversions.
One standard way to record computations in the realm of very small finite groups is to use a string of different characters (digits or letters) to denote the permutation which transforms the sorted elements in the top row into the matching elements of the bottom row. Both row are placed between parentheses. Juxtaposition of two such notation indicates the composition of the functions so denoted, with the usual convention that the rightmost function is to be applied first (composition isn't commutative):
A cyclic permutation of n elements is denoted by a sequence between parentheses. The image of an element is the element to its right (the last element is mapped back to the first one). There are n equivalent ways to denote such a permutation, since there's a free choice of which element is written first in the list. Equivalent notations are equated:
(1 4 5 3 2) = (4 5 3 2 1)
A cycle is a permutation of n elements which is a cyclic permutation of m of those elements (m ≤ n) which leaves the others unchanged. Two or more cycles are said to be disjoint when operate on different elements (each cycle applies only to elements which are left unchanged by the others). Any permutation can be decomposed as a composition of disjoint cycles in a unique way (up to the order of those cycles, which is irrelevant since disjoint cycles commute). Our previous example entail cycles which do not commute because they're not disjoints. namely:
(3 4) (2 3) = (2 4 3) (2 3) (3 4) = (2 3 4)
A cycle of order 2 (a 2-cycle) is called a switch or a transposition. It's useful to know that the signature of a cycle of order n is (-1)n+1.
Cycle Notation for Permutations (12:36) by Liliana De Castro (Socratica, 2018年10月15日).
Arthur Cayley (1821-1895) observed that a group G is always isomorphic to a subgroup of Sym(G).
Proof : In the multiplicative group G, we associate to an element a the bijection T(a) which sends an element x to ax . T is an injective homomorphism (i.e., a monomorphism) from G to Sym(G), which is called the regular representation of G.
T(a) o T(b) = T(a b)
So, any finite group of order n is isomorphic to a subgroup of Sn . QED
Numericana : Yoneda lemma (Category theory)
To any element a of G is associated a special type of automorphism, called an inner automorphism (French: automorphisme intérieur ) defined as follows ( fa is called conjugation by a ).
" x, fa(x) = a x a-1 [ Note that fa o fb = fab ]
Under function composition, inner automorphisms form a normal subgroup (see proof later in this section) denoted Inn(G), of the group of the automorphisms on G, denoted Aut(G) (itself a subgroup of Sym(G), the symmetric group on G). Conjugation by a is the identity function just if a belongs to the center of G. Consequently:
Inn(G) is isomorphic to the quotient of G by its center.
Note that a subgroup H of G which is mapped onto itself by any inner automorphism is a normal subgroup (also called invariant subgroup).
More generally, two subgroups of G are said to be conjugates of each other when there is an inner isomorphism between them.
The above claim that Inn(G) is a normal subgroup of Aut(G) is established by showing that conjugation by any automorphism g of an inner automorphism (conjugation by a) yields another inner automorphism. That can be proved in a single line:
" x, g o fa o g-1 (x) = g ( a g-1(x) a-1 ) = g(a) x g(a)-1 = fg(a) (x)
The outer automorphism group of the group G is defined as the quotient of its group of automorphisms by its group of inner automorphisms :
Out(G) = Aut(G) / Inn(G)
Unfortunately, the elements of Out(G) are known as outer automorphisms although they're not "automorphisms" at all !
A group G is said to be centerless when its center is trivial, which is to say that only the identity element commutes with every element.
A complete group is a centerless group whose only automorphisms are the inner ones. (Equivalently, it's a group whose center and outer automorphism group are trivial.)
If a group G is complete, it's isomorphic to Aut(G) (its automorphisms). However, the converse need not be true (one counterexample is D4 ).
"Classification of small complete groups" (2010) in Math Exchange and Math Overflow.
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Orbit-counting theorem (Cauchy-Frobenius lemma) | Redfield-Pólya theorem (1927,1937)
Two elements x and y of a group G are said to be conjugates when there's an inner automorphism from one to the other, that is, when there's an element a of G such that ax = ya.
So defined, conjugacy is an equivalence relation (it's reflexive, symmetric and transitive). The conjugacy class of an element x is the set of all elements of G which are conjugate to it. Every element is in one and only one of those classes (equivalence classes always form such a partition).
If x is in the center of G, denoted Z(G), then the conjugacy class of x is simply {x} (a set of only one element). More generally, we would establish that the number of elements that are conjugate to x is equal to the index in G of the centralizer C of {x}. That number is usually denoted [ G : C ].
Tallying the conjugacy classes with more than one element by assigning each a different index i, we obtain the so-called conjugacy class formula :
| G | = | Z(G) | + å i [ G : Ci ]
The second term is an empty sum (equal to zero) when G is commutative.
{e} and G are trivially always normal subgroups of G. The group G is said to be simple when its only normal subgroups are those two.
Just like 1 isn't said to be prime, the trivial group {e} isn't called "simple".
Simple groups (8:52) by Liliana De Castro (Socratica, 2018年01月10日).
The commutator [x,y] of two elements of the multiplicative group G is:
[x,y] = x y x-1 y-1 = x y (y x)-1
The set of all commutators isn't necessarily a subgroup. What's called the derived subgroup (or commutator subgroup) is the subgroup they generate (i.e., the smallest subgroup which includes all commutators).
A group is said to be perfect when it's equal to its derived subgroup. In particular, a group which contains only commutators is perfect. That's so for all finite non-abelian simple groups, as was first conjectured by Øystein Ore (1899-1968) in 1951. Ore's conjecture was proved in 2010, using the classification of finite simple groups.
The derived subgroup of a group is a normal subgroup, as the following identity demonstrates (since the set of commutators is thus shown to be stable under any inner automorphism, so is the subgroup they generate).
a [x,y] a-1 = [ axa-1, aya-1 ]
G' is also the smallest normal subgroup of G whose quotient group in G is abelian (i.e., commutative). The group G/G' is known as the abelianization of G (it's the largest abelian quotient in G).
The derived subgroup of any abelian group is the trivial subgroup.
The derived subgroup of the symmetric group Sn is the alternating group An. The derived subgroup of the alternating group is itself: A'n = An.
The derived subgroup of the Quaternion group is {+1,-1}.
It's the sequence where the n+1 st term is the derived subgroup of the n-th one (starting with the whole group when n = 0).
Commutator Subgroup | Commutator | Derived series | Perfect core
The direct product of two groups G and H is the group obtained by using for the cartesian product G ´ H independent operations on the components:
(g,h) (g',h') = ( g h , g'h' )
The term direct sum is used for the same concept with additive notations:
(g,h) + (g',h') = ( g+h , g'+h' )
Similar rules can be used for cartesian products of any number of monoids.
The concept extends naturally to direct sums (or direct products ) of infinitely many monoids. Such direct sums are usually understood to be finitely restricted (by considering just the elements having only a finite number of components that differ from the relevant neutral element).
This assumption is always made in the case of vector spaces (only finitely many components are nonzero in the resulting structure) and it's prudent to clearly distinguish between the two possibilities for infinite cartesian products endowed with component-wise operations.
For example, the fundamental theorem of arithmetic provides a standard isomorphism between the multiplicative monoid of the positive integers and the finitely restricted direct sum of infinitely many copies of the nonnegative integers (each such copy being associated with a prime number). Using standard notations, this can be expressed as:
Note that the set appearing in the right-hand-side of the above is countable, because of the parenthesized exponent which indicates a finite restriction in the above sense. A lack of parentheses around the exponent would denote an uncountable set which is rarely investigated, if ever (that beast includes elements idenfified with products of infinitely many coprime integers).
Thus, if n is the k-th power of a prime, the number of non-isomorphic abelian groups is equal to the number p(k) of partitions of k.
More generally, if the prime factorization of n is q1k1 q2k2... qmkm then the number of non-isomorphic abelian groups of order n is equal to:
Abel ( n ) = p( k1 ) p( k2 ) ... p( km )
That's a multiplicative function of n. which depends only on its prime signature (A000688).
By trying only the first 61, we see that the only partition numbers which divide 1000000 are p(1) = 1, p(2) = 2 and p(4) = 5. Therefore, there are exactly 1000000 distinct abelian groups of order n if and only if the factorization of n consists of:
The smallest example is a 35-digit integer:
(2 . 3 . 5 . 7 . 11 . 13)4 (17 . 19 . 23 . 29 . 31 . 37)2 = 4.96597898...1034
Exactly one million abelian groups of order n? (Mathematics Stack Exchange, 2014年05月08日).
Burnside theorem (1904) | William Burnside (1852-1927)
If f is a homomorphism from a group H to Aut(G), the semi-direct product of G and H with respect to f is the group denoted G ´f H consisting of the cartesian product G ´ H with the multiplication :
(x,a) (y,b) = ( x f (a) (y) , ab )
When f is the trivial homomorphism (i.e., f (a) is the identity of G for any a) this semi-direct product is just the direct product of G and H.
When H is equal to Aut(G) we may use the identity of Aut(G) as the homorphism f appearing in the above definition and define the holomorph Hol(G) as the semi-direct product of G and Aut(G) in which :
(x,a) (y,b) = ( x a(y) , ab )
Holomorph of a group
|
Semidirect products (inner or outer)
The
Automorphisms of the Automorph of a Finite Abelian Group (1956) by
William H. Mills (1921-2007?).
Additive notations (using the symbol "+" for the operator) are often used for commutative groups (abelian groups). Groups isomorphic to the group Cn = (Z/nZ, +) of residues modulo n are called cyclic groups.
The above is a special case of the notation A/I which denotes a ring obtained as a quotient of a ring by one of its ideals. As such it's a structure endowed with two operators. The single-operator additive group of that structure is properly denoted (A/I,+). The expression (A/I,+,x) is pleonastic. The notation Zp is for something else.Cyclic Group C5
All groups of prime order are cyclic (as Lagrange's Theorem implies that the subgroup generated by a nonneutral element is equal to the entire group). The same is true for groups whose order is a cyclic number (i.e., an integer coprime to its Euler totient). That result is due to William Burnside.
The smallest noncyclic groups are thus of order 4 and 6. The Klein group is the noncyclic group of order 4. The smallest noncommutative group is the following group S3 = D3 (the 6 symmetries of an equilateral triangle).
Klein GroupThe dihedral group Dn consists of the 2n symmetries of a regular n-gon (n rotations, n flips).
August KekuleWhen he proposed the cyclic structure of benzene in 1865, August Kekulé (1829-1896) thought that the C6H6 molecule had trigonal symmetry (expressed by the order-6 group D3 tabulated above) because of his vision that single and double bonds were alternating along the carbon ring. The currently accepted symmetry for the benzene molecule is the hexagonal group D6 (of order 12) with 3 of the binding electrons in a delocalized orbital covering the whole ring.
There are 5 groups of order 8. Three are abelian : C8 and the two direct sums C2+C4 and C2+C2+C2 (the additive group of the field of order 8). The other two groups of order 8 are noncommutative, namely the dihedral group D4 (the symmetries of a square) and the quaternion group Q8 :
On October 16, 1843, the fundamental equations below (which imply the given multiplication table) occurred at once to Hamilton as he was crossing Brougham Bridge (Broom Bridge) in Dublin. He carved them into the stone of the bridge (the original carving is gone but a plaque celebrates this act of "mathematical vandalism").
The real line combined with an oriented 3-dimensional Euclidean space of orthonormal basis (i,j,k) forms the quaternions, a 4-dimensional normed division algebra similar to 2-dimensional complex numbers, except multiplication is not commutative:
This is how the 3-dimensional "dot product" and "cross product" were invented, well before the generalized idea of a vector became commonplace.
The above quaternionic units can be used to build a Dirac operator D (yielding the opposite of the Laplacian D when applied twice):
D = i ¶ / ¶x + j ¶ / ¶y + k ¶ / ¶z
The Laplacian remains the same in two systems of coordinates (a.ka. reference frames) obtained from each other by rigid rotation.
History of Quaternions (41:08)
by Kathy Joseph (Kathy Loves Physics & History, 2023年01月30日).
Fantastic Quaternions (12:24)
by James Grime (Numberphile, 2016年01月18日).
Visualizing quaternions, with stereographic projection (31:50)
Grant Sanderson (3Blue1Brown, 2018年09月06日).
The multiplicative group generated by the four gamma matrices a,b,c,d is of order 32. It consists of 16 disjoint pairs of elements which are (additive) opposites of each other. With one element of each such pairs, we form a basis for the algebra of dimension 16 generated by the 4 gamma matrices.
The group clearly contains the identity matrix of dimension four (I) and the product e = abcd. Introducing e allows every element of the group to be uniquely represented by a sign (+ 1 or -1) along with a signed product of at most two factors among the five elementary elements a,b,c,d,e. With zero such factors, we have 2 elements (+I and -I), with one factor we have 10 elements (a,b,c,d,e and their opposites) and with 2 distinct factors, we obtain 20 = 2×C (5,2) elements. The grand total is indeed 32.
Among the 32 elements of the gamma group, we find:
This algebra of dimension 16 is known as the spacetime algebra Cl (1,3) which is just the Clifford algebra of dimension 4 with Minkowski metric.
Pseudoscalars (Grade 4) commute only with scalars or bivectors (Grade 2). The bivectors and the scalars form the centralizer of the pseudoscalars.
The above table doesn't depend on Dirac's representation of a,b,c,d in terms of 4×4 matrices. It can be entirely constructed from the pairwise anticommutativity of a,b,c,d and the following relations. Therefore, an isomorphic group is entirely specified by a choice of a basis of four mutually anticommutative elements verifying these:
a2 = I , b2 = c2 = d2 = -I and the definition e = abcd
The last relation implies that e2 = -I and also that e anticommutes with the other four. Thus, a is special (it's the only single-letter element which squares to unity) but b,c,d,e are placed on an equal footing.
Here's one remarkable identity:
det ( ta + xb + yc + zd + ue) = ( t 2 - x 2 - y 2 - z 2 - u 2) 2
Let f be an automorphism. f (a) must be of order 2 and is therefore, up to a change of sign, an element of {a,ab,ac,ad,ae}. For the three distinct images of b,c,d to anticommute with f (a) and with each other, they must belong to {b,c,d,e} up to sign. Conversely, if those conditions are met, the images of a,b,c,d generate the whole group, as the above table can be constructed using only the rules for combining single letters. Thus, we have 5 choices for f (a) and 4×3×2 choices for the other three letters, knowing that we may then pick any choice of four signs among 16 possibilities. Therefore:
Dirac's Gamma group has 5! 24 = 120 × 16 = 1920 automorphisms.
This is a centerless group G isomorphic to Aut(G) but not to Inn(G). A nice example of an incomplete group isomorphic to its automorphisms.
The dihedral group D4 can be represented as the group of the 8 symmetries of a square, with vertices numbered clockwise 1,2,3,4. It's generated by :
If there was an automorphism swapping an odd number of the three pairs (B,D), (E,G) and (F,H) then we could combine it with one of the four inner automorphisms to obtain some automorphism f leaving (A,B,C,D) invariant and swapping either (E,G) or (F,H). Neither is possible, since:
Therefore, any other automorphism must involve sending at least one element of the three aforementioned pairs to an element of another.
Any automorphism must leave invariant A (the identity) and C (the only other element with a square root). Likewise, the order-4 elements, B and D, must be invariant or transform into each other.
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Aut (D4 ) , the group of automorphisms of D4 , is isomorphic to D4.
One of the 8 isomorphisms between D4 and Aut (D4 )The 9 proper subgroups of D4 are abelian. Seven of them are cyclic (one of order 1, five of order 2, one of order 4) and two are Klein groups.
The octic group may also be represented as a group of 2 by 2 matrices:
Automorphisms of the Dihedral Groups (1942) by George Abram Miller (1863-1951).
If the integer n is coprime with its Euler totient f(n), then there's only one group of order n (the cyclic group). This applies to the following values of n: 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51... (A003277). This result is attributed to William Burnside (1852-1927) and those numbers are known as cyclic numbers.
A divisor of a Carmichael number is necessarily odd and cyclic. Around 1980, I made the conjecture that the converse is true; any odd cyclic number seems to divide at least one Carmichael number (if the conjecture is true, it divides infinitely many of them, since a cyclic number has infinitely many cyclic multiples). In 2007, Joe Crump and myself proved this to hold for cyclic numbers below 10000. We're not attempting to gather more numerical evidence at this time...
For noncyclic orders (A060679) here's the number of distinct groups:
Number of groups of order n (A000001)g(n) = 2 if n is either the square of a prime or a squarefree number with only one of its prime factors congruent to 1 modulo another (A054395). The following table gives, for each m, the numbers n for which g(n) = m.
Smallest n for which there are exactly m groups of order n
(A046057):
1, 4, 75, 28, 8, 42, 375, 510, 308, 90, 140, 88, 56, 16, 24, 100, 675...
The convention would be to have a zero term to indicate that there's no such n but it's conjectured that this never happens.
Groups of order 2n (A000679) | The Small Groups Library
The finite simple abelian groups are just the cyclic groups of prime order.
The classification of noncommutative finite simple groups is much tougher... Arguably, the final classification effort started with the 1963 publication of a 255-page proof of the Odd Order Theorem (or Feit-Thompson theorem) which implies that all noncommutative simple finite groups are of even order:
Solvability of Groups of Odd Order
by John G. Thompson (1932-) and Walter Feit (1930-2004).
Pacific Journal of Mathematics 13 (1963) 775-1029.
The classification was declared complete in 1982, despite pending gaps... This was the result of a tremendous collective effort, spanning decades. A key figure in this accomplishment was Daniel Gorenstein (1923-1992).
Unless it's one of the 27 sporadic groups presented below (including the Tits Group, often dubiously tallied with twisted Chevalley groups) a finite simple group must belong to one of the following 18 countable families:
An(q) = PSL (n+1, Fq )
Chevalley groups are named after Claude Chevalley (1909-1984) who was the youngest founder of the Bourbaki group in 1935.
In 1955, Chevalley found a uniform way to describe Lie groups over arbitrary fields. With finite fields, this led to what J.H. Conway (1937-2020) and others have called untwisted Chevalley groups (they're listed first in the above table, with unsuperscripted symbols).
The twisted Chevalley groups (denoted by superscripted symbols) result from two modifications of Chevalley's approach. One was proposed in 1959 by Robert Steinberg (1922-2014). The other (1960-1961) is due to Michio Suzuki (1926-1998) and Rimhak Ree (1922-2005).
The above highlighted entry 2F4(2 2m+1 ) is simple only for positive values of m. For m=0, this group is not simple but it has a simple normal subgroup of index 2 and order 17971200 (its derived subgroup) which is known as the Tits Group, and is best classified among sporadic groups.
Classification of Finite Simple Groups
|
List of Finite Simple Groups
Jordan-Hölder theorem
|
Camille Jordan (1838-1922; X1855)
|
Otto Hölder (1859-1937)
Simple Groups (8:52)
by Liliana De Castro (Socratica, 2017年01月10日).
20 of these are related to the largest and most famous of them all, the Fischer-Griess Monster. Six other sporadic groups ( highlighted ) unrelated to the Monster are known as oddments or pariahs.
The 27th sporadic group is, arguably, the aforementioned Tits Group.
The Mathieu group M21 doesn't belong to the above list. It's simple but can't be considered sporadic because it's isomorphic to PSL(3,4):
M21 = PSL(3,4) = PSL(3,F4 ) = A2 (4)
The Fischer-Griess Monster Group is also known as Fischer's Monster or the Monster Group. It was predicted independently by Bernd Fischer and Robert L. Griess in 1973. At first, Griess dubbed it the Friendly Giant and constructed it explicitely in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.
The Leech Lattice is the densest packing of 24-dimensional hyperspheres (each touches 196560 others). Its automorphisms feature a center of order two. Modulo that center, they form the Conway Group (Co1).
Simon P. Norton gave a construction of the group proposed by Koichiro Harada (now called the Harada-Norton group). Norton also proposed the monstruous moonshine conjecture with his advisor, John H. Conway.
The Higman-Sims Group (HS) is named after Donald G. Higman and Charles C. Sims, who described it jointly in 1968. It's a subgroup of index 2 in the group of automorphisms of the Higman-Sims graph (the strongly-regular graph with 100 nodes of degree 22, where adjacent nodes have no common neighbors and nonadjacent nodes have 6 common neighbors).
The Hall-Janko Group (HJ) is named after Marshall Hall, Jr. (1910-1990) and Zvonimir Janko (1932-). It's a subgroup of index 2 in the automorphisms of the Hall-Wales graph constructed in 1968 by David Wales, as the strongly-regular graph with 100 nodes of degree 36, where adjacent nodes have 14 common neighbors and nonadjacent nodes have 12 (also called Hall-Janko graph).
The modern quest for a complete list of sporadic groups was launched by the discovery of the first of the Janko Groups (J1) by Zvonimir Janko, in 1965.
The first sporadic groups (M11 , M12 , M22 , M23 , M24 ) are subgroups of M24 discovered between 1860 and 1873 by Emile Mathieu (1835-1890; X1854). Georg Frobenius (1849-1917) proved M12 to be simple in 1904.
The Mathematicians Involved
|
Monstrous Moonshine Theory (Wikipedia)
Video :
Finite groups, Yesterday and Today
by Jean-Pierre Serre (Harvard, 2015年04月24日).
An element of finite order is called a torsion element. If the identity is the only such element, the group G is said to be torsion-free.
A torsion element whose order divides k is called a k-torsion.
On the other hand, a torsion group (also called a periodic group) is a group consisting only of torsion elements (which is to say that all elements have finite orders). All finite groups are periodic (i.e., Tor(G) = G). If the orders of the elements in a periodic group are bounded, then they have a least common multiple n and the group is said to be of exponent n.
One example of an infinite finitely-generated torsion group was given in 1964, by Evgeny Golod (1935-2018) and Igor Shafarevich (1923-2017).
Burnside problem (1902) | William Burnside (1852-1927)
GL(n,K) is the group of invertible n by n matrices with entries in a field K.
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All finite groups are linear.
Compact groups...
Lie groups...
Faithful representations (isomorphisms).
Irreducible representations do not allow any nontrivial
proper invariant subspace.
Shur's lemma
|
Issai Schur (1875-1941)
Representation Theory Basics
by Robert Donley (2011年03月08日).
The tangent space to a Lie group is a Lie algebra. The converse is true in finitely many dimensions but there are Lie algebras with infinitely many dimensions which cannot be realized as the tangent space to a Lie group. The earliest counterexample is due to the bourbakist Adrien Douady (1935-2006).
The classical groups tabulated below are subgroups of the group GL(n,K) of invertible n by n matrices with entries in the field K.
When K isn't specifed, the field of real numbers (R) is understood, except that the field of complex numbers (C) underlies the groups denoted U(n) and SU(n) (note, however, that the "dimension" listed is always the real dimension, which is twice the complex dimension whenever applicable).
A subgroup of GL(n,K) is called a linear representation (or simply a representation) of any group it happens to be isomorphic to.
A* denotes the adjoint of the square matrix A (namely, the "conjugate transpose" of a complex matrix, or simply the transpose of a real matrix).
A matrix is said to be unimodular if its determinant is 1. In the symbol of a group, the letter "S" (for special) says that its elements are unimodular.
A notation like GL(Kn) may also be used instead of GL(n,K). This has the great advantage of being consistent with more general symbols like GL(V) which apply to a vector space V whose dimension may be infinite.
On the other hand, when a finite field is used, GL(n,GF(q)) may be denoted GL(n,q). A similar convention holds for all the symbols tabulated above. For example, the first type of Chevalley groups is PSL(n,q) = An(q).
There's no risk of confusion with notations like O(3,1) as used below, which refer to a real vector space metrically endowed with 3 spacelike dimensions and 1 timelike dimension, since we've yet to conceive several dimensions of time and rarely consider a field of one element.
Classical groups | "The Classical groups" (1939) by Hermann Weyl (1885-1955)
With K = R (or C) the above classical groups are examples of Lie groups.
Lie Algebra of a Lie group
Lie groups and their Lie algebras (1:43:11)
by Frederic P. Schuller (IAS, 2015年09月21日).
Traditionally, the projective group is the quotient of the general linear group (i.e., the group of all square matrices of a given dimension over a given field) modulo the scalar group (i.e., the diagonal matrices).
The term is also used as a qualifier to denote the quotients nodulo the scalar group of some subgroups of the general linear group.
By extension, the qualifier projective can even be used to denote the quotient of any group modulo its own center. (See modular group.)
The qualifier "projective" is inherited from the name given to the rules of geometrical perspective, first devised by Renaissance artists. In their drawings, they mapped every point (P) of three-dimensional Euclidean space to the unique point (M) of a planar canvas intersecting the straight line (OP) drawn from that point to the eye of the observer (O). In such a mapping, a horizontal plane is mapped onto a half-plane of the canvas which ends on a straight line representing the horizon (supposedly "at infinity").
In one of the greatest leaps of imagination ever made by the human mind, the geometers of the nineteen century realized that this artistic rendering was just a special case of the above and they would eventually turn projective geometry into a very fruitful independent field of study. Once called higher geometry, that became a revered part or higher learning before being all but forgotten...
An homographic transformation f (also called a Möbius transformation or a fractional linear transformation) sends a complex number z to:
It's a [bijective] transformation of the projective line (the complex plane plus a single "infinity" point ¥ beyond its horizon, so to speak). The image of ¥ is a/c (or ¥ if c = 0 ). The image of -d/c (or ¥ if c = 0 ) is ¥.
Automorphic functions (originally dubbed "Fuchsian functions" by Poincaré, around 1884) are meromorphic functions (i.e., ratios of two holomorphic functions; analytic functions of a complex variable) which are invariant under a countable infinity of Möbius transformations).
Moebius Transformations Revealed (a great video) by Douglas N. Arnold & Jonathan Rogness
The locution gamma group is best reserved for something else.
The modular group consists of all 2 by 2 square matrices with integer elements (in Z) and unit determinant (that's what special means) when considered modulo the center {I,-I} (that's what projective means).
That last specification merely states that a matrix and its opposite are equivalent representations of the same element of the modular group.
The modular group G has the following presentation:
G = < S, T | S2 , (ST)3>
G is a discrete subgroup of the Möbius group, represented as follows:
The modular group was first studied in detail, for its own sake, by Richard Dedekind and Felix Klein as part of the Erlangen program (1872). The closely related elliptic functions (introduced by Lagrange in 1785) had already been studied quite extensively by Abel (1827-1828) and Jacobi (1829) who shared the grand prix of the French Academy of Sciences for that work, in 1830 (after Abel's death).
An interesting source of examples in the modular group is provided by the successive convergents obtained by truncating the continued fraction expansion of a number, because the following relation is naturally satisfied:
Pn+1 Qn - Qn+1 Pn = (-1)n
Elliptic functions,
modular forms,
Hecke theory,
etc.
&
theta functions
by
Ben Brubaker (MIT, 2008).
Generating
the modular group (Malik Younsi, 2010)
&
Euclidean algorithm (Qiaochu Yuan, 2008).
The modular group and words in its two generators
by Giedrius Alkauskas (2016年04月10日)
A265434
The
Modular Group ... subgroups McCreary, Murphy, Carter.
The Mathematica Journal, 9, 3 (2005).
Geometry and Groups
by T. Keith Carne, University of Cambridge (2012).
The Modular Group and its Actions
by A. Muhammed Uludag, with Appendix by Hakan Ayral (2013).
The Minkowski Question Mark and
the Modular Group by Linas Vepstas (2014).
Wikipedia :
Modular group
|
Moduli spaces
In the Euclidean plane, a cubic curve without singular points is called an elliptic curve. That same term is also commonly used to denote the cartesian equation of such a curve or the wonderful group structure its points can be endowed with, as described below. Elliptic curves can be considered over various fields (complex numbers, rationals, p-adic numbers, finite fields).
Elliptic curves over finite fields allow elliptic-curve cryptography (ECC) which was invented in 1985 and has been widely used since 2004.
[画像: Come back later, we're still working on this one... ]
In 1901, Poincaré had asked whether the rational points of a curve of genus 1 are finitely generated. 21 years later, Mordell settled that for elliptic curves:
An elliptic-curve's rational points form a finitely-generated abelian group.
For an elliptic curve E, this is denoted E(Q)
Elliptic curves
|
Group law of an elliptic curve
|
Addition theorems
|
Finitely generated abelian group
EEC: Elliptic-curve cryptography (1985)
|
Neal Koblitz (1948-)
|
Victor S. Miller (1947-)
Birch and Swinnerton-Dyer conjecture about the rank of an elliptic curve.
In the Euclidean plane, let's apply the geometric definition of sums on an elliptic curve to the degenerate cubic consisting of a circle of unit diameter and a straight line at a distance d from its center.
When at least one point is on the circle, the geometric construction of the sum of two points presents no difficulty. On the other hand, if both of the points A and B are on the line, their sum C = A+B is not immediately clear. To construct it, we may consider any auxiliary point V on the circle and use the following identity, involving three sums of the previous kind:
A + B = ( (A+V) + B) - V
For convenience, we choose V on the axis of symmetry of the figure, so that V = -V, in which case we have a symmetrical defining relation:
A + B = (A+V) + (B+V)
If A' is the mirror-image of A+V (with respect to the horizontal axis of symmetry)
then the law introduced in the non-degenerate case says that
A' is at the intersection of the circle and the AV line. Likewise,
the image B' of B+V is the intersection of BV with the circle.
A+B is on the mirror-image of the line joining A+V and B+V,
which is the line A'B'. So, A+B is at the intersection of A'B'
with our basic vertical line, as shown in the figure at left.
[画像: Come back later, we're still working on this one... ]
If we're concerned with number theory, we choose any rational value for k. Otherwise, we remark that the above equation encodes a group structure on the real line in one of three different ways, modulo some rescaling:
Moreover, the limiting case when k tends to infinity can be construed as ordinary multiplication of the reciprocals of nonzero numbers. Of course, the (nonzero) rational numbers are not finitely generated under this law, because there are infinitely many prime numbers.
More generally, we may consider any continuous monotonous function f from negative infinity to positive infinity and define an abelian group law over the real numbers by:
x o y = f ( f -1 (x) + f -1 (y) )
Our previous discussion is a special case of that if we choose f to be either the trigonometric tangent or the hyperbolic tangent. The former for a line which doesn't intersect the basic circle, the latter for a line which does.
Relativistic addition of parallel velocities | Broken-calculator puzzle
An amenable group is a locally compact topological group whose elements leave invariant some kind of averaging on bounded functions.
The English word was coined in 1949 by Mahlon M. Day as a pun ("a-mean-able") to translate the German term originally used by Von Neumann in 1929 (messbar = measurable). The French use either the English term or the (better) word moyennable.
The three Thompson groups F, T and V are also called vagabond groups, chameleon groups or just chameleons (the latter term was coined by Matt Brin in 1994). They have unusual properties which have made them counterexamples to several conjectures in group theory.
F is not simple but its derived group is. T and V are simple. T was the first known example of a finitely-presented infinite simple group.
F can be defined as the subgroup of the piecewise-linear automorphisms of the interval [0,1] consisting of all functions f such that:
[画像: Come back later, we're still working on this one... ]
Thompson groups (F, T and V)
|
Richard J. Thompson
(1933-2015?)
Ph.D. 1979)
|
Matthew G. Brin
The Higman-Thompson groups
by Peter Cameron (2016年06月05日)
|
Graham Higman (1917-2008)
The many faces of Thompson's Group F by
John Meier (2004-09)
What is Thompson's Group? by
J.W. Cannon & W.J. Floyd (2011-08)
Presentations of Thompson's group V by permutations
by Collin Bleak & Martyn Quick (2015年11月06日).
An introduction to Thompson's group F (and T) for physicists (33:32)
by Tobias Osborne (2016年01月27日).
The chameleon groups of Richards J. Thompson
by Matthew G. Brin (IHES, 84, pp. 5-33, 1996).
[画像: Come back later, we're still working on this one... ]
Let M be a sandpile for which there's a sandpile Z such that M+Z = M. Then, Z is a zero (i.e. it's a neutral element for addition) over the set of all sandpiles of the form X+M, since:
(X+M) + Z = X + (M+Z) = X+M
[画像: Come back later, we're still working on this one... ]
Diffusion limited aggregation (DLA, 1981)
|
Thomas A. Witten, Jr.
(1944-)
|
Leonard M. Sander
(1941-).
Abelian sandpiles (1987)
|
Per Bak (1948-2002)
|
Tang Chao (1958-)
|
Kurt Wiesenfeld (1958-)
The Abelian Sandpile and Related Models
by Deepak Dhar (1998年10月22日).
The Abelian sandpile; a mathematical introduction
by R. Meester, F. Redig, D. Znamenski (2001年06月12日).
What is a Sandpile?
by Lionel Levine & James Propp (Notices of the AMS, 57, 8, pp. 976-979, Sept. 2010).
Primer for the Algebraic Geometry of Sandpiles
by D. Perkinson, J. Perlman, & J. Wilmes (2012年01月04日).
Sandpile Fractal Generation (1:16:29)
by Harrison Lambeth (2017年02月01日).
Convergence of the Abelian sandpile (57:48)
by Charles Smart (Microsoft Research, 2016年04月16日).
Self-Organized Criticality (10:01)
by Andrew Hoffman (2015年05月12日).
Sandpiles (24:09)
by Luis David Garcia-Puente
(Numberphile, 2017年01月13日).
The Lorentz Group O(3,1) is isomorphic to SL(2,C) and consists of all 4 by 4 real matrices A such that A* h A = 1, where h is the metric matrix for three dimensions of space and one dimension of time.
The O(3,1) group has 4 connected components. Those components are pairwise homeomorphic and are not simply connected :
SO+(3,1) T[ SO+(3,1) ] P[ SO+(3,1) ] PT[ SO+(3,1) ]
SO+(3,1) is the (6-dimensional) Restricted Lorentz Group consisting of the elements of the Lorentz Group O(3,1) which preserve the direction of time and the orientation of space (boosts and 3D rotations). In the above, T and P denote a reversal of time and an inversion of space (the latter could be either a mirror symmetry about a plane or a symmetry about a point).
The symbol SO(3,1) would denote the "Special Lorentz Group", the subgroup of the matrices of O(3,1) with determinant one (which is a disconnected "half" of O(3,1), not a connected "quarter" of it).
The Poincaré Group ISO+(3,1) is the 10-dimensional inhomogeneous group of noninverting isometries for 3 dimensions of space and one dimension of time. It consists of transformations mapping x to Lx+a , where L belongs to the above Restricted Lorentz Group SO+(3,1) and a is some 4-vector.
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Wikipedia :
Wigner's classification
|
Wigner's theorem (1931)
|
Eugene Wigner (1902-1995;
Nobel 1963)
Baker-Campbell-Hausdorff formula
God does arithmetic.
Carl Friedrich Gauss (1777-1855)
In spite of their respective successes, General Relativity and the Standard Model are known to be imperfect theories, incompatible with each other. The ultimate laws of physics (if they exist) could only incorporate those two as approximations applicable to specific experimental domains (like Newtonian mechanics approximates Special Relativity for low speeds).
Nobody knows (yet) exactly what symmetries the ultimate laws of nature should have, but we may ponder the groups of local symmetries underlying modern mathematical theories of the 4 known physical interactions:
Maxwell's unification of electricity and magnetism into electromagnetism has been ultimately construed as the discovery that electrodynamics is invariant under local phase transformations, with the simple structure of U(1). The classical quantity associated with that symmetry (by Noether's theorem ) is simply electric charge.
Quantum electrodynamics (QED) describes electromagnetism as a quantum field. It became the basic paradigm for all subsequent quantum theories of fundamental physical interactions. QED describes how photons "mediate" the force between electrons (or any other charged particles).
The electroweak theory is a satisfying unification of electromagnetism and weak interactions under the symmetries of the direct product SU(2)´U(1). It was devised in 1967 by Steven Weinberg (1933-) and Abdus Salam (1926-1996) building on earlier work of Sheldon Glashow (1932-). The three men shared the 1979 Nobel prize for this. The group SU(2) is isomorphic to 3-dimensional rotations. The broken electroweak symmetry translates into 4 vector bosons: g (the photon) Z0, W+ and W-.
Broken: In mathematical physics, a symmetry is said to be broken when symmetrical equations have an asymmetrical solution.
The theory of strong interactions is known as quantum chromodynamics (QCD). It's based on an unbroken SU(3) local symmetry, dubbed color symmetry because of a superficial similarity with the rules of color vision (whereby 3 primary colors may combine to create colorlessness). QCD describes how gluons mediate the strong force between quarks (or anything else with color charge, including gluons themselves). There are 8 different types of gluons, corresponding to the 8 dimensions of SU(3). In this context, SU(3) is often denoted SUc(3). "C" stands for color.
As described by Albert Einstein's General Theory of Relativity, gravity's local symmetry is that of the Poincaré group, which preserves spacetime intervals, as well as the direction of time and the orientation of space. The Poincaré group is 10-dimensional. However, a gauge field (the graviton) is associated only with the 4 dimensions of spacetime translations. Suspiciously, no such particle or field is associated with the 6 dimensions corresponding to Lorentz symmetries (3 dimensions for spatial rotations and 3 dimensions for Lorentz boosts).
The so-called Standard Model of particle physicists describes both strong and electroweak interactions in a theoretical framework whose symmetries are those of the group SU(2)´U(1)´SUc(3), which has 12 dimensions.
The model depends on several parameters, adjusted to fit experimental data but otherwise unexplained. Different local symmetries would impose different restrictions, for better or for worse. One classical group possessing more dimensions of symmetry (24) than the Standard Model is SU(5).
The correct local symmetry of "strong-electroweak" interactions would still not determine the masses of the vector bosons involved (particles of spin 1) unless more is known about the way such a symmetry is broken.
A key aspect of particle physics which is based on a broken symmetry is the classification of elementary particles into three generations of flavors.
A mind-boggling supersymmetry across different spins (SUSY) seems required of any quantum theory designed to include gravity in a fully unified quantum theory "of everything": Supergravity, Superstrings, etc.
In 2010, Sir Michael Atiyah (1929-2019) remarked that the known physical symmetries occur naturally in the Tits-Freudenthal magic square pertaining to associative hypercomplex numbers obtained through the Cayley-Dickson construct. He speculated that the introduction of the fourth (nonassociative) hypercomplex division algebra (the octonions) is somehow related to gravity. He admonished younger investigators to consider this possibility, which would give a beautiful role to all exceptional Lie groups.
Yang-Mills theory (1954)
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Weak interactions
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Strong interactions
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String theory
Pauli matrices for SU(2)
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Gell-Mann matrices for SU(3)
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Generalizations
Grand Unified Theories : SU(5), SO(10) ...
Yang-Mills Theories (3:21)
by Murray Gell-Mann (1998).
Mirror Symmetry & Geometric Langlands (1:11:21)
by Ed Witten (2012年10月18日).
Les Symétries de l'univers (French, 15:34)
by Alessandro Roussel (ScienceClic, 2021年02月06日).
Dirac's belt trick, Topology, and Spin ½ (59:42)
by Noah Miller (2021年08月23日).
The name cosmic Galois group was coined by Pierre Cartier around 1998, as he shared the optimism of Fields medalist Maxim Kontsevich (1964-) in the following words:
La parenté de plus en plus manifeste entre le groupe de Grothendieck-Teichmüller (GT) d'une part, et le groupe de renormalisation de la Théorie Quantique des Champs n'est sans doute que la première manifestation d'un groupe de symétrie des constantes fondamentales de la physique, une espèce de groupe de Galois cosmique !
The subject caught the attention of several other people connected with the IHES. One comprehensive introduction appears in the work (2004) of Alain Connes (1947-) and Matilde Marcolli (1969-).
[画像: Come back later, we're still working on this one... ]
Grothendieck-Teichmüller group (Drinfeld, 1990)
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Absolute Galois group (of the rationals)
Vladimir Drinfeld (1954-)
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Alexander Grothendieck (1928-1914)
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Oswald Teichmüller (1913-1943)
nLab
Cosmic Galois Group
by Pierre Cartier (Lectorium, Euler Institute, 2013年10月21日).
