Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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4
votes
2answers
185 views

Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?

I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
1
vote
0answers
72 views

If $G=\prod_{p\in P}\mathbb Z_p$ then $\frac{G}{tG}$ is divisible.

I want to show that: If $G=\prod_{p\in P}\mathbb Z_p$, wherein $P$ is the set of all primes, then $\frac{G}{tG}$ is divisible. I know that $tG$ is not a direct summand and if $x\in G$ wants to ...
1
vote
1answer
84 views

$F/H$ has an element of infinite order?

Here is a problem: If $F$ is a free abelian group of rank $n$ and $H$ is a subgroup of rank $k<n$, then $F/H$ has an element of infinite order. What I did: I assume $F=\langle ...
2
votes
1answer
99 views

$\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$

I want to prove that: If $F$ is a free abelian group of rank $n$, then $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$ with ...
2
votes
1answer
85 views

Verifing some properties about $G$

I have the following problem$^*$: Prove that the group $G$ having generators and relations respectively $$X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$$ is an infinite ...
1
vote
1answer
58 views

$G$ is torsion-free group then $G/\langle X\rangle$ is torsion

Honestly, I have been thinking on this problem for hours but couldn't find a way: Let $G$ is torsion-free group and $X$ is a maximal independent subset, then $G/\langle X\rangle$ is torsion. I ...
1
vote
1answer
108 views

Divisible abelian $q$-group of finite rank

What does "finite rank" mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
1
vote
1answer
69 views

Exact sequence of abelian groups, property transfers

We had the statement that with an exact sequence of multiplicatively written abelian groups $U \mapsto V \mapsto W \mapsto X \mapsto Y$ and in $U$, $V$, $X$, $Y$ every group element has a unique ...
2
votes
2answers
142 views

Prove $H \times G$ is commutative iff $H, G$ are commutative

This is another proof question I am asking about, can someone give me tips on how to answer these questions? My question says: "Let $H,G$ be arbitrary groups. Prove that $H \times G$ is commutative ...
1
vote
2answers
833 views

determining number of subgroups

If $G$ is an abelian group of order 72, do we know how many subgroups of order 8 it has? Just because it's a divisor doesn't mean that there is a subgroup of that size. But I'm wrong. Why?
1
vote
1answer
259 views

Groups of the same order that are nonisomorphic

I'm reading A First Course in Abstract Algebra by Fraleigh and I've reached a point where I feel like I'm supposed to have understood something more from the chapter than what is actually stated. I've ...
4
votes
1answer
158 views

Smallest pure subgroup containing a fixed subgroup

I will ask a slightly more precise question then in the title. Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
3answers
284 views

Splitting exact sequences of finite abelian groups

I would like to find a condition for an exact sequence of abelian groups $$ 0\to H\to G\to K\to 0 $$ to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
4
votes
1answer
442 views

An indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable

I'd like to find an indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable, and I'm running out of ideas: The only indecomposable $\mathbf{Z}$-modules I know are $\mathbf{Z}$, ...
3
votes
3answers
201 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
0
votes
2answers
75 views

Elementary Question about Torsion Subgroups

Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$. Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime. Denote by $G[a]$ resp. $G[b]$ the $a$-resp. ...
4
votes
1answer
232 views

When abelian group is divisible?

By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$. Let $A$ be abelian group with no proper subgroups of finite index. How can I prove ...
3
votes
1answer
162 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
5
votes
1answer
363 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...
0
votes
0answers
59 views

Abelian group problem with three consecutive intergers [duplicate]

Possible Duplicate: Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian If $G$ is a group such that $ (a \circ b)^i = a^i \circ b^i $ for ...
5
votes
2answers
225 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
6
votes
6answers
5k views

Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
8
votes
3answers
190 views

Tensor-commutative abelian groups

Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$. The first question is somewhat vague: Question 1. ...
2
votes
1answer
64 views

The $p^k$-rank of a subgroup is no greater than the $p^k$-rank of the group.

Recently I was given a handout containing (roughly) the following text: Let $A$ be a finite abelian group, and $p^k$ a prime power. The $p^k$-rank of $A$ is defined to be ...
4
votes
1answer
257 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
3
votes
1answer
283 views

exponent of an abelian group

Let $p$ be a prime. Let $H_{i}, i=1,...,n$ be normal subgroups of a finite group $G$. I want to prove the following: If $G/H_{i}$, $i=1,...,n$ are abelian groups of exponent dividing $p−1$, then $G/N$ ...
4
votes
1answer
293 views

Why do characters on a subgroup extend to the whole group?

As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally ...
1
vote
0answers
69 views

Pontrjagin's Lemma and an application

I would appreciate any kind of help on the following issue: On page 114 of Rotman's "Homological Algebra", exercise 3.4 reads: 1) (Pontrjagin) If an abelian group $A$ is countable, torsion-free ...
3
votes
1answer
145 views

CEP for Abelian groups and lattices

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta \in \operatorname{Con} B$ there is a $\phi \in \operatorname{Con} A$ such that $\theta = \phi \cap (B\times ...
-3
votes
2answers
751 views

Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
3
votes
1answer
480 views

Torsion-free quotient group of an abelian group

Let $G$ be an abelian group, and let $H\leq G$. Prove that if $G/H$ is torsion free, then $H$ contains the torsion group of $G$. Proof: Let $x\neq1$ be an element in the torsion group. Thus there ...
3
votes
3answers
272 views

Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element

Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$: $$\begin{aligned} a \odot b=a+b+3\end{aligned}$$ Check if $(\mathbb Z_7, \odot)$ is a group and ...
0
votes
1answer
244 views

To check given group is abelian. [duplicate]

Possible Duplicate: If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian? I want to show that group $G$ is abelian (i.e. $ab=ba$) if $a^{3}=e, \forall a\in G.$ I am trying so ...
2
votes
1answer
349 views

Group Extensions of Finite Abelian Groups

Given a short exact sequence of finite abelian groups, is it possible to classify what groups can show up in the middle based on the kernel and the cokernel? I'm hoping the answer is much easier (than ...
5
votes
0answers
102 views

Possible subgroups of $\mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$

$G \cong \mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$ $H\leq G$ so that $G/H \cong \mathbb{Z}/3^2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z} $ Find all possible ...
9
votes
1answer
191 views

For which $n$, $G$ is abelian?

My question is: For Which natural numbers $n$, a finite group $G$ of order $n$ is an abelian group? Obviouslyو for $n≤4$ and when $n$ is a prime number, we have $G$ is abelian. Can we consider ...
4
votes
3answers
566 views

Quotient of two free abelian groups of the same rank is finite?

Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B ]$ (the index of $B$ in $A$) is finite. For ...
4
votes
1answer
269 views

Is this the free abelian group functor?

Let $\mathbb{Z}(.) : \mathbf{Set} \to \mathbf{Ab}$ be the functor that assigns to any set $S$ the set of maps $\mathbb{Z}(S) := \{ z: S \to \mathbb{Z} \; | \; z(s)=0 \mbox{ for almost all } s \in S ...
5
votes
2answers
372 views

If $G$ is finite and abelian, then every subgroup of $G$ is characteristic if and only if $G$ is cyclic

Suppose $G$ is finite and abelian. Show that every subgroup of $G$ is characteristic if and only if $G$ is cyclic. I have the 'if' part so far: If $G$ is cyclic, then $G = \langle g \rangle $ ...
5
votes
1answer
319 views

Freiman homomorphism on generating set

I got stuck in an exercise from Tao and Vu's book Additive Combinatorics. It is ex. 5.3.4. on page 226. In the following let (Z,+) and (W,+) be two abelian groups and let A $\subset$ Z and B ...
9
votes
1answer
451 views

Does the splitting lemma hold without the axiom of choice?

In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} ...
3
votes
1answer
215 views

Non-Boolean group with every element of order two

Let $G$ be a group (not necessarily finite) such every element of $G$ has order 2. Every such group is abelian [1]. Clearly, every Boolean algebra $B$ is a group of this type, when equipped with the ...
1
vote
1answer
352 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...
1
vote
2answers
206 views

Elements of order $6$ in an Abelian group of order $360$

Let $A$ be a finite abelian group of order $360$ which does not contain any elements of order $12$ or $18$. How many elements of order $6$ does $A$ contain? I've got that $A$ is $C_2 \times C_6 ...
1
vote
1answer
358 views

Free abelian group $F$ has a subgroup of index $n$?

Suppose that we have a free abelian group $F$. How can it be proved that $F$ has a subgroup of index $n$ which $n≥1$? Honestly, according to the Theorems, I just know that if we take $X$ as a base ...
17
votes
1answer
1k views

Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?

I think the answer is yes. Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then ...
2
votes
1answer
706 views

Determining the Smith Normal Form

Consider the integral matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 ...
6
votes
1answer
1k views

An abelian group of order 100

The first part of the problem asks you to prove that an abelian group $G$ with order $100$ must contain an element of order $10$. For this part, I use Sylow theorem to list possiblities for $H$ and ...
6
votes
1answer
2k views

Converse of Lagrange's theorem for abelian groups

I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...
1
vote
5answers
230 views

How to prove a group has a basis with exactly one element?

I am struggling with the following question. Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix} a \\[0.3em] b ...