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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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Irreducible complex representations of some abelian Lie groups

I wanted to classify all irreducible complex representations of the following basic abelian Lie groups: $\mathbb{S}^1$ the circle in the complex plane, $\mathbb{R}_{>0}$ the positive real numbers, $...
Don Abbondio's user avatar
-1 votes
0 answers
44 views

Isomorphism Between quotient groups of two groups linked by a surjective group homomorphism [closed]

In an exercise I am trying to solve, we are given two abelian groups $(G;*)$ and $(H;\square)$ and are asked to prove the equivalence of the following statements: There exists a surjective group ...
TinCan's user avatar
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2 votes
1 answer
18 views

Problem of finding the invariant factors and elementary divisors of a quotient group.

$\mathbf{The \ Problem \ is}:$ Let $G:= \Bbb{Z}_9\times \Bbb{Z}_9\times \Bbb{Z}_9$ and $H:= \langle(6,6,6)\rangle.$ Find the invariant factors and elementary divisors of the quotient group $G/H.$ $\...
Rabi Kumar Chakraborty's user avatar
2 votes
1 answer
110 views

When is the abelian groups homomorphism $g \mapsto 2g$ injective?

This is possibly an easy question. Let $(G,+)$ be a countable abelian group. Under what circumstances is the homomorphism $\phi: G \mapsto 2G:=\{g+g: g\in G\}$, defined by $\phi(g)=g+g$ injective? I ...
User's user avatar
  • 654
5 votes
0 answers
55 views

Uniqueness of module structure

Question: Let $E$ be a torsion abelian group. Prove that $E$ has exactly one $\hat{\mathbb{Z}}$-module structure, and that the scalar multiplication $\hat{\mathbb{Z}} \times E \rightarrow E$ defining ...
ByteBlitzer's user avatar
6 votes
3 answers
281 views

If $ (ab)^2=b^2 a^2 , \forall a , b\in G$ , then $G$ is necessarily abelian?

Let $G$ be a group such that $(ab)^2=b^2a^2 \forall a,b\in G $ then $G$ is abelian? I tried to find a counterexample, but nothing came up. Hence I tried to prove it. We have $(aba^-)^2=ab^2a^- $ and ...
user-492177's user avatar
  • 2,557
0 votes
1 answer
22 views

Baer's definition of $S$-characteristic subgroup

I am reading a part of Baer's paper Groups without proper isomorphic quotients, Bull. AMS,50 (1944) 267-278. In that paper, he also discusses groups without proper isomorphic subgroups, which he calls ...
Chris Leary's user avatar
  • 3,003
6 votes
1 answer
104 views

If $F$ is an endofunctor on the category of abelian groups $G \mapsto G_{tors}$, then $R^1F(A) \cong A \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z}$

I am trying to find all right derived functors of the functor $F: Ab \to Ab$ defined as an endofunctor on the category of abelian groups $G \mapsto G_{tors}$. I have proven that $R^iF = 0$ for all $i &...
Squirrel-Power's user avatar
0 votes
1 answer
56 views

Abelian groups that can be extended with non abelian solvable groups [closed]

Let $H$ be a non-trivial finite abelian group. Is it true (in some cases) that there exist a non-abelian solvable group $G$ and a normal abelian subgroup $K$ of $G$ such that $G/K$ isomorphic to $H$? ...
Rozina Ali's user avatar
3 votes
1 answer
68 views

Proving fundamental group of some topological space is non-abelian [closed]

Let $X$ be the unit circle $S^{1}$ with two line segments $\{0\} \times [-1, 1]$ and $[-1,1] \times \{0\}$ as in the figure below. Prove that the fundamental group $\pi_{1}(X, (0,1))$ is non-abelian. ...
JLGL's user avatar
  • 627
4 votes
1 answer
46 views

For a reduced abelian $p$-group $G$, does $P$ being finite imply $G$ is finite?

Let $G$ be a reduced abelian $p$-group, and let $P$ be the subgroup of elements of order $p$. If $P$ is finite, is $G$ necessarily also finite? Recall that a group $G$ is said to be reduced if it ...
rea_burn42's user avatar
-2 votes
0 answers
24 views

primary decomposition of a factor group

Decompose $\mathbb{Z}^{*}_{17 \cdot 257 \cdot 65537} / \{\pm 1\}$ into primary groups. I know that $(\mathbb{Z} / 17 \cdot 257 \cdot 65537 \mathbb{Z})^* / \{ \pm 1 \} \cong (C_{17} \times C_{257} \...
ABlack's user avatar
  • 578
2 votes
0 answers
35 views

Pre-Sheaf on locally compact abelian group

Let $G$ be a locally compact abelian group. We can define a pre-sheaf $\mathcal{F}$ on G by $$\mathcal{F}(U) := \widehat{\langle U\rangle}$$ for any open set $U$, where $\widehat{\langle U\rangle}$ ...
Pedro Lourenço's user avatar
-1 votes
1 answer
102 views

Character table for a covering group of $\mathbb{Z}_n \times \mathbb{Z}_m$

I’m considering the group $G = \mathbb{Z}_n \times \mathbb{Z}_m$ and its covering group $$G^* = \langle \alpha, \beta, a|\alpha a = a\alpha, \beta a = a\beta, a^p = 1, \alpha^n = 1, \beta^m = 1, \...
slowspider's user avatar
  • 1,057
1 vote
0 answers
48 views

Extension topology

I am reading a paper by Goldman and Sah on extension topology, but I am uncertain about the meaning of the sentences. Paraphrasing the paper: Let $X$ be an abelian group and $M$ be a subgroup. ...
YSA's user avatar
  • 151
2 votes
1 answer
61 views

Abelian group with maximal number of subgroups of order 18

Among all the abelian groups of order 21600 find one that has the maximal number of subgroups of order 18. Decompose this group into primary groups. Here's my attempt: $21600 = 3^3 2^5 5^2$, so the ...
ABlack's user avatar
  • 578
0 votes
1 answer
55 views

If $A$ and $B$ are abelian groups, is $(A \times B)/A$ isomorphic to $B$?

If $A$ and $B$ are abelian groups, is $(A \times B)/(A\times \{0_B\})$ isomorphic to $B$? If not, do we need additional conditions?
Blue2357's user avatar
0 votes
0 answers
23 views

Locally compact group whose unitary irreducible reps are one dimensional

It's known that if G is a finite group such that all its irreducible unitary representations are one-dimensional, then G is abelian. This uses the fact that the left regular representation decomposes ...
Pedro Lourenço's user avatar
-1 votes
1 answer
41 views

Powers of non-subgroup element [closed]

Let $G$ be finite abelian group and $H$ its proper subgroup of non-prime order $\omega$. For $g \in G, g \notin H$ is it possible that $g^t \in H$, where $t$ is divisor of $\omega$ ?
Vitaliy Volovyk's user avatar
6 votes
1 answer
80 views

Is a subgroup in $\mathbb Q^n$ satisfying some condition free abelian?

Consider $\mathbb Q^n$ as an abelian group under addtion, $G\subset \mathbb Q^n$ is a subgroup, such that: For any $v\in\mathbb Q^n$, the subgroup $(\mathbb Q\cdot v)\cap G$ is either isomorphic to $0$...
cybcat's user avatar
  • 481
1 vote
2 answers
48 views

Question over decomposition of $\mathbb{Z}_{mn}$

If $m<n\in\mathbb{N}$ and $(m,n)=1$, then there is a natural isomorphism $h: \mathbb{Z}_{mn}\to \mathbb{Z}_m \times \mathbb{Z}_n$. But I'm a little confused about what happens when multiplying $m$ ...
user760's user avatar
  • 1,500
1 vote
1 answer
92 views

what should be the group $B$?

Here is the exact sequence of abelian groups I am studying: $$0 \to \mathbb Z/2 \to B \to \mathbb Z/2 \to 0 $$ Can I say that $B \cong \mathbb Z/2$ or $B \cong \mathbb Z/2 \oplus \mathbb Z/2$? Is $B \...
Emptymind's user avatar
  • 2,069
1 vote
0 answers
17 views

In finite abelian group, extend 2-cocycle from subgroup

Let $G$ be a finite abelian group, $K\leq G$ a subgroup and $[\alpha] \in \operatorname{H}^2(K,\mathbb{C}^{\times})$. Is it always true that one can extend $\alpha$ to $G$, i.e. find $[\beta] \in \...
david's user avatar
  • 45
4 votes
1 answer
76 views

Finitely generated abelian groups with the same finite quotients

Let $\Gamma$ and $\Delta$ be two finitely generated abelian groups. Therefore, by the classification theorem of finitely generated abelian groups, we can assume that $\Gamma \cong \mathbb{Z}^r \oplus ...
Aron's user avatar
  • 263
2 votes
0 answers
24 views

$G$ abelian $p$-group and $1 \neq A$ $p'$-subgroup of $Aut(G)$. Then $A$ acts faithfully on $\Omega_1(G)$.

Let $G$ be an abelian $p$-group and let $1 \neq A$ be a $p'$-subgroup of automorphisms of $G$. Then $A$ acts faithfully on $\Omega_1(G)$. Proof: Let $1 \neq a \in A$. According to the Lemma we ...
Stippinator's user avatar
9 votes
1 answer
178 views

Is $\text{Hom}(A,\mathbb{Z})$ a product of free abelian groups for all abelian groups $A$?

Let $A$ be an abelian group, and consider the abelian group $\text{Hom}(A,\mathbb{Z})$ of homomorphisms from $A$ to $\mathbb{Z}$. What can be said about this group? Since $\mathbb{Z}$ is torsion-free, ...
Lukas Lewark's user avatar
0 votes
1 answer
69 views

Symmetric difference as a group law

Let $D$ be any set and define $\mathcal{P}(D)=\{A:A\subseteq D\}$ to be the power set of $D$. For any $A,B\in\mathcal{P}(D)$, define the symmetric difference operation $$A*B=(A\setminus B)\cup (B\...
clathratus's user avatar
  • 17.3k
0 votes
0 answers
16 views

Topological tensor product

Let $A$ and $B$ be topological abelian groups. Consider the following universal property for a pair $(P, \pi)$, where $P$ is a topological abelian group and $\pi: A \times B \to P$ is a jointly ...
Smiley1000's user avatar
  • 1,472
2 votes
1 answer
42 views

KL is an abelian extension implies K and L are abelian extensions

Prove of give a counterexample: If $KL$ is abelian then $K$ and $L$ are abelian. I have proven the converse, that is if $K$ and $L$ are abelian then $KL$ is abelian. I have a feeling the above doens'...
JLGL's user avatar
  • 627
2 votes
1 answer
86 views

$C(X, A) \otimes C(X, B) \cong C(X, A \otimes B)$?

Consider the category $\mathsf{Ab}$ of abelian groups and $\mathsf{TopAb}$ of topological abelian groups. Given a topological space $X$ and a topological abelian group $A$, define the abelian group $C(...
Smiley1000's user avatar
  • 1,472
2 votes
1 answer
139 views

Counting maximal subgroups of $\mathbb{Z}_m^n$

Let  $$ \mathbb{Z}_m=\mathbb{Z} / m\mathbb{Z} $$ How many maximal subgroups does  $$ \mathbb{Z}_m^n=\underbrace{\mathbb{Z}_m \times \mathbb{Z}_m \times \cdots \times \mathbb{Z}_m}_n $$  have? (m need ...
tys's user avatar
  • 163
0 votes
1 answer
55 views

$G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle.$ Proof by induction on $|G|$

I have a question on a proof found in our lecture book on group theory from Gernot Stroth. I do not understand what here is meant by "Induction on $|G|$", which makes it very hard to grasp ...
Stippinator's user avatar
1 vote
1 answer
73 views

Explicitly finding a finite abelian group from a presentation, $\langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$

I have the following group presentation: $ G= \langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle $ It is clear that $G$ is a finite abelian group. I am interested in knowing what exactly $G$ ...
eyp's user avatar
  • 127
0 votes
1 answer
45 views

The number of direct sum of elementary abelian 2-groups

Let $G=(Z_2)^n$, I want to know the number of direct sum of $G$($G=H \oplus K$) or a fine upper bound. For $G=Z_2 \oplus Z_2$, I have calculated that all of its direct sum decomposition is as follows: ...
zeyu hao's user avatar
  • 347
4 votes
1 answer
110 views

The category of abelian groups with quasi-homomorphisms

Let $A$ and $B$ be abelian groups. Say that a map $f: A \to B$ is a quasi-homomorphism if there exists a finite $D \subseteq B$ such that $$\forall a_1, a_2 \in A: f(a_1 + a_2) - f(a_1) - f(a_2) \in D$...
Smiley1000's user avatar
  • 1,472
4 votes
2 answers
90 views

How to show that $\mathrm{Tor}_{2}^{\mathbb{Z}}=0$

Since $\mathrm{Tor}$ functor preserve direct limit and any Abelian group is direct limit of its finitely generated subgroups, and by the fundamental theorem of f.g. abelian groups, it reduces to the ...
ymx ddl's user avatar
  • 144
1 vote
0 answers
38 views

Quick question about correction to a typo in an example about the directed system of abelian groups?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Example We consider the category of Abelian groups, and let $(A_i),(f_{ij}), ...
Seth's user avatar
  • 3,545
0 votes
1 answer
49 views

Endomorphisms of a direct sum of abelian groups.

Let $I$ be a non-empty set and let $(A_i,+_i,0_i)$ for $i\in I$ be an abelian group, such that $Hom(A_i,A_j)=0$ for $i,j\in I, i\neq j$. Then $f$ is an endomorphism of a group $A=\bigoplus_{i\in I}{...
alpha's user avatar
  • 218
1 vote
1 answer
55 views

If $G$ is an additive group, $u,v$ endomorphisms, then if $h(x)= x-u(v(x))$ is onto then $f(x)= x-v(u(x))$ is onto

Let $G$ be an additive group, and $u,v\colon G \to G$ homomorphisms. I have to show that the map $f\colon G \to G$, $f(x)=x-v(u(x))$ is surjective if the map $h\colon G\to G$, $h(x)=x-u(v(x))$ is ...
NoetherNerd's user avatar
3 votes
1 answer
63 views

In an abelian group, is a certain inequality involving products of cardinalities of sum and differences of finite subsets (sumsets) true or not?

For any two finite subsets $A,B$ of an abelian group, does the following ineqality hold? $$|A+B|^2 |A-B|^2 \geq |A+A||A-A||B+B||B-B| \ $$ I’m interested in finding out if there are sumset inequalities ...
Teddy Astor's user avatar
1 vote
1 answer
76 views

Skew-symmetric non-degenerate bicharacters over abelian groups

I'm struggling to solve a problem in group cohomology theory which could seem immediate to some more expert mathematicians here. Suppose to have a non-degenerate, skew-symmetric bicharacter $b\colon G\...
skewfield's user avatar
  • 123
1 vote
0 answers
24 views

metabelian groups can be represented by matrices, do we know the exact representation?

We know that every finitely generated metabelian group can be represented by matrices (e.g. see https://link.springer.com/article/10.1007/BF02219822 ). I am particularly interested in how finitely ...
ghc1997's user avatar
  • 1,511
4 votes
1 answer
61 views

Prove that $\sum_{i=1}^{k} \lambda_i f(g_i x) \geq 0$ holds for all $x \in G$, then $\sum_{i=1}^{k} \lambda_i \geq 0.$

Problem statement: Let $f(x) \geq 0$ be a nonzero, bounded, real function on an Abelian group $G$, $g_1, \ldots, g_k$ are given elements of $G$, and $\lambda_1, \ldots, \lambda_k$ are real numbers. ...
Martin.s's user avatar
1 vote
2 answers
79 views

Let $G$ be a finite abelian group of order n. How many distinct group homomorphisms are there from $G$ to $R/Z$?

First, is this question well-defined? That is, the homomorphisms are determined by the order n. Since $R/Z$ is the multiplicative unit circle, all finite subgroups are cyclic, so the image of the ...
Halulu's user avatar
  • 29
-2 votes
1 answer
43 views

Is every element of a finite abelian group contained in a cyclic factor?

Suppose $A$ is a non-cyclic finite abelian group and $a\in A$. Can we always find $A_1,A_2\leq A$ such that $a\in A_1$ and $A=A_1\oplus A_2$? Equivalently, is it true that given a finite abelian $A$ ...
tomasz's user avatar
  • 35.8k
2 votes
1 answer
114 views

Proof of the universal property of free abelian groups

Let $S$ be a set. The group with presentation $(S, R)$ , where $R = \{\ [s , t] \mid s,t\in S\ \}$ is called the free abelian group on $S$ -- denote it by $A (S)$ . Prove that $A (S)$ has the ...
Dian Wei's user avatar
  • 351
0 votes
0 answers
22 views

The subgroup and direct summand of an abelian group

Let $A=\mathbb{Z}\oplus\mathbb{Z}_3.$ I want to find all subgroups of $A$ as well as all direct summand. For the subgroup, let $H=H_{free}\oplus H_{tor}$. And consider the inclusion: $\iota_1:H_{free}\...
Zeta's user avatar
  • 155
4 votes
1 answer
79 views

An abelian group of order 35 must be cyclic [duplicate]

I'm reading through Contemporary Abstract Algebra by Joseph A. Galian and I stumbled upon the following problem: Suppose $G$ is an Abelian group of order $35$ and every element of $G$ satisfies $x^{...
Selim Bamri's user avatar
0 votes
1 answer
35 views

Condition for an abelian group to be equipped with $\mathbb{Z}/n\mathbb{Z}$ -module structure [closed]

I have this algebra question : Let $n$ be any integer strictly greater than $1$, and $(G,+)$ an abelian group such that for all $x$ in G, $nx=0$. Show that $(G,+)$ can be equipped with $\mathbb{Z}/n\...
Charbel Doumit's user avatar
0 votes
1 answer
120 views

Show that a group is commutative if and only if another operation exists with certain properties

Prove that a group $(G, \cdot )$ is commutative if and only if we can define on G another operation "$\circ$" such that: i) $x \circ x = e$, for every $x \in G$ ii) $x \circ (y \circ z)=(x ...
Maish Petronas's user avatar

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