Groups

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Definition

x(yz) = (xy)z, xe = x, xx' = e (Common definition)

x(yz) = (xy)z, ex = x, x'x = e (Another common definition)

x(yz) = (xy)z + for all x there exists a unique x' such that xx'x = x

(w((x'w)'z))((yz)'y) = x [McCune1993] shows by exhaustive computer search that this is the shortest single identity in these operations capable of defining groups.

Examples

• Groups of small order
• Any abelian group
• Symmetric group S_n of all permutations of n objects, under composition of mappings. They have order n! and presentations

  s_1,..,s_{n-1}; s_i^2 = 1, s_i s_j = s_j s_i if i < j-1 s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} 
   

• The dihedral groups D_n, of order 2n, with presentations
• The braid groups B_n, of infinite order, with presentations

  s_1,..,s_{n-1}; s_i s_j = s_j s_i if i < j-1 s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} 
   

• GL(n,F) = invertible n x n matrices over any field F
• SL(n,F) = n x n matrices of determinant 1 over any field F
• SO_n = rotations of Euclidean n-space
• Automatic groups

Structure

Representation

Decision problems

Identity problem: Solvable

Word problem: Unsolvable [Novikov][Boone] Here is a group presentation with 10 generators and 27 relations which has unsolvable word problem. It is given in D.J. Collins, "A simple presentation of a group with unsolvable word problem," Illinois J. Math 30 (1986), 230-234, based on work of G.S. Céjtin and V.V. Borisov:

 <a,b,c,d,e,p,q,r,t,k;
 p^{10}a = ap, p^{10}b =bp, p^{10}c = cp, p^{10}d = dp, p^{10}e = ep,
 aq^{10} = qa, bq^{10} =qb, cq^{10} = qc, dq^{10} = qd, eq^{10} = qe,
 ra=ar, rb=br, rc =cr, rd=dr, re=er,
 pacqr = rpcaq, p^2adq^2r = rp^2daq^2,
 p^3bcq^3r = rp^3cbq^3, p^4bdq^4r = rp^4dbq^4,
 p^5ceq^5r = rp^5ecaq^5, p^6deq^6r = rp^6edbq^6,
 p^7cdcq^7r = rp^7cdceq^7, p^8ca^3q^8r = rp^8a^3q^8,
 p^9da^3q^9r = rp^9a^3q^9,
 pt = tp, qt = tq, 
 a^{-3}ta^3k = ka^{-3}ta^3> (This last one is misprinted in Collins)
 

Spectra and growth

Finite spectrum: (numbers of groups of orders <= 100, from J.A. Gallian, "Contemporary Abstract Algebra", p. 290)

 1, 1, 1, 2, 1, 2, 1, 5, 2, 2,
 1, 5, 1, 2, 1, 14, 1, 5, 1, 2,
 2, 2, 1, 15, 2, 2, 5, 4, 1, 4,
 1, 51, 1, 2, 1, 14, 1, 2, 2, 14,
 1, 6, 1, 4, 2, 2, 1, 52, 2, 5,
 1, 5, 1, 15, 2, 13, 2, 2, 1, 13,
 1, 2, 4, 267, 1, 4, 1, 5, 1, 4,
 1, 50, 1, 2, 3, 4, 1, 6 1, 52,
 15, 2, 1, 15, 1, 2, 1, 12, 1, 10,
 1, 4, 2, 2, 1, 230, 1, 5, 2, 16,...
 

Free spectrum:

Growth series: (1+z)/(1-(2r-1)z) for the free group on r generators.

History/Importance

They're very important and have a long history.

References

Too many to give here.

Subsystems

Abelian groups, solvable groups, nilpotent groups

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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu