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Questions tagged [abelian-groups]

For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.

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5 votes
1 answer
283 views

This question is distantly related to the following MathStack post: How "big" can the center of a finite perfect group be? The above post and its answers comment on the size of the center of ...
0 votes
0 answers
33 views

The motivation for this problem is that it is a component in proving the Freyd-Mitchell embedding theorem. Let $F$ be a functor from a small abelian category $C$ to the category of abelian groups $\...
2 votes
2 answers
211 views

I'm implementing a type conversion routine for a programming language. Under arithmetic context, there can be 4 types: null, ...
2 votes
2 answers
196 views

This is a second followup question to this question I asked a couple of days ago (here is the first followup question). After resolving the issues I raised in both of the linked questions I proceeded ...
2 votes
1 answer
74 views

This is a followup question to this question I asked yesterday (which by the way I was almost able to figure out entirely), in which I tried to prove that: Let $R$ be a commutative ring with identity ...
2 votes
1 answer
137 views

I have the following homework problem: Compute $H^\bullet(Y),ドル where $Y$ is the universal cover of the mapping torus space $X$ of a degree-2ドル$ map $S^2 \to S^2$. (Hint: $H^2(Y) \cong \Bbb Z_{(2)}/\...
4 votes
1 answer
136 views

Define a double field to be a set $D$ with three binary operations, $+,\times,\Delta,ドル all commutative, with identity elements 0,1,ドル\omega$ respectively, such that $$(D,+,\times)$$ is a field and $$(...
2 votes
0 answers
63 views

One of the exercises in the textbook I am using, W. R. Scott's Group Theory, asks me to prove that "[a] factor [i.e. quotient] group of a direct sum of cyclic groups is not necessarily a direct ...
0 votes
0 answers
22 views

Let $G$ be an abelian $p$-group. It is well-known that $G$ has a basic subgroup $B$. The group $B$ has several properties: (a) $B=\bigoplus_{n=1}^{\infty}B_n,ドル with $B_n=\bigoplus_{\kappa_n}\mathbb{Z}(...
4 votes
0 answers
97 views

Let $\operatorname{Top}$ be the category of topological spaces with continuous maps between them. Let $\operatorname{Unif}$ be the category of uniform spaces with uniformly continuous maps between ...
2 votes
1 answer
113 views

I'm taking a course in algebra and the teacher presented me with the following problem: Let G be a group such that the derived subgroup $G'=[G,G]$ is a direct factor, i.e., there exists a subgroup $K\...
0 votes
1 answer
113 views

According to definition of a field, If F is an abelian group under + and F - {identity element of (F,+)} is an abelian group under • and it shows distributive property then F is a field. Can there be ...
2 votes
2 answers
204 views

Let $S^1 := \{z \in \mathbb{C} \;|\; \left\lvert z\right\rvert = 1\}$ — the standard unit circle which is also a topological group. Surely, there are continuous homomorphisms $z \mapsto z^n$ for each $...
5 votes
1 answer
145 views

Let $(G,*)$ be a group , $a \in G$. Prove that if $xa^3=ax^3$ for all $x \in G$ then $G$ is Abelian. If $a$ is the identity then we get $x^2=e$ and then is easy to show that $G$ is Abelian. Otherwise, ...
5 votes
1 answer
147 views

Let $G$ be an abelian group (not necessarily finitely generated), and let $H$ be a finitely generated subgroup of $G$. Does there exist a finitely generated subgroup $H'$ of $G,ドル with $H\subseteq H',ドル ...

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