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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3
votes
1answer
137 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
734 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
468 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
271 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
240 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
338 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
189 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
555 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
268 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
359 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
317 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
213 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
343 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
203 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
354 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
692 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 ...
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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 ...
3
votes
1answer
90 views

exact sequence of finite abelian groups which are squares

Let $0 \to A \to B \to C \to 0$ be an exact sequence of finite abelian groups. Assume that $B$ and $C$ is a square (i.e. there are groups $D,E$ such that $B \cong D^2$, $C \cong E^2$). Does this imply ...
2
votes
2answers
333 views

Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$

Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$ Since $\mathbb{C}[x]$ s noetherian we have that $\mathbb{C}[x]/(x^2)$ is too. And thus finitely generated ...
0
votes
1answer
109 views

Finding subgroups of index 2 of $G = \prod\limits_{i=1}^\infty \mathbb{Z}_n$

I looked at this question and its answer. The answer uses the fact that every vector space has a basis, so there are uncountable subgroups of index 2 if $n=p$ where $p$ is prime. Are there ...
0
votes
2answers
113 views

Torsion subgroup quotient

Let G be an abelian group, $T$ the torsion subgroup of $G$. If $G/T$ is torsion-free, then $T$ and $G/T$ must be disjoint. $G=T \bigoplus G/T$ implies this as well. I don't understand why they are ...
2
votes
1answer
116 views

Trouble with decomposition of groups of order 2009

A question says: prove or prove or disprove that there are only 2 non-isomorphic abelian groups of order 2009. I think that it is true because... I split up $2009$ into $7 \times 7 \times 41$ and so ...
5
votes
2answers
2k views

Proving that a subgroup of a finitely generated abelian group is finitely generated

A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated. I would say that for a finitely generated abelian ...
3
votes
1answer
211 views

Finding subgroups of index 2

Let $G = \prod_{i=1}^\infty \mathbb{Z}_2$ with addition mod 2. I am trying to find subgroups of index 2. I see that taking the entire space and removing all sequences which have a 1 in a certain ...
11
votes
3answers
1k views

A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$

A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ such that $$B \cong A \oplus C$$ although the sequence does ...
4
votes
1answer
286 views

how to show that a group is elementarily equivalent to the additive group of integers

Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers? I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
3
votes
1answer
175 views

Simplify the category of finite abelian groups

Consider the category $\mathsf{FinAb}$ of finite abelian groups. The structure theorem tells us that we can write down a skeleton for this category (a set of representatives for the isomorphism ...
0
votes
1answer
91 views

What are the Subgroups of the abelian group $\mathbb{Z}_5$?

How can I find the subgroups of abelian group $\mathbb{Z}_5$? From Lagrange's theorem, the size of the subgroup should divide 5 in this case. So the size of the subgroup should be 1 or 5 ...
1
vote
0answers
71 views

Normal and Abelian groups? [duplicate]

Possible Duplicate: A, B subgroups of G, B/A abelian. Show that BN/AN is abelian. Let A, B, N be subgroups of a group G such that A $\triangleleft$ B and B/A is Abelian. Also suppose N ...
0
votes
2answers
633 views

How can I prove an abelian group is not free?

How can I prove a given abelian group; such as $\mathbb{Z}_4$ with addition mod 4, is not a free group? Should I consider all the subsets of the given group and prove any of them cannot be a basis? ...
0
votes
1answer
405 views

Show that fiber products exist in the category of abelian groups.

Show that fiber products exist in the category of abelian groups. In fact, If $X, Y$ are abelian groups with homomorphisms $f: X \to Z$ and $g: Y \to Z$ show that $X \times_z Y$ is the set of all ...
6
votes
1answer
5k views

Computing the Smith Normal Form

This question is related to the Smith Normal Form of Matrices: Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $R :=$ $$ \begin{bmatrix} -6 & 111 & ...
5
votes
2answers
846 views

If a group is $3$-abelian and $5$-abelian, then it is abelian

In a group $(Z,*)$, $(a*b)^{5}=a^{5}*b^{5},\forall a,b\in Z$ and $(a*b)^{3}=a^{3}*b^{3}$ then prove that $Z$ is abelian. I know that for three consecutive integer if $(a*b)^{i}=a^{i}*b^{i},\forall ...
1
vote
1answer
152 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
1
vote
2answers
1k views

Show that a group of order $5$ is always abelian.

Show that a group of order $5$ is always abelian. I know that if the binary operation $*$ is defined on a group $G$ is commutative (i.e. $a*b=b*a\ \forall a,b\in G$),\ then G is called a ...
3
votes
1answer
203 views

If $G$ has a normal subgroup of order 2 and infinite cyclic quotient, $G$ is abelian?

Assume that $G$ has a normal subgroup $H$ of order $2$ (isomorphic to $Z_{2}$) and $G/H$ is infinite cyclic (which indicates that $G$ is also infinite order). The target here is to prove that $G$ is ...
4
votes
1answer
348 views

abelian transitive subgroups

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? For instance: Let $G$ be a an abelian transitive subgroup of the symmetric group ...
2
votes
1answer
148 views

Find the structure of $\mathbb{Z}[\sqrt[3]{2}]/(4+\sqrt[3]{4})$

Let $A=\mathbb{Z}[\sqrt[3]{2}]$ and $I=(4+\sqrt[3]{2^2})$. Elements in $A$ have the form $a\cdot 1+b\cdot 2^{\frac{1}{3}}+c\cdot 2^{\frac{2}{3}} \Rightarrow$ elements in $I$ have the form $$ (a\cdot ...
0
votes
1answer
108 views

Finitely generated abelian group has a regular normal form

Prove that every finitely generated abelian group admits a regular normal form. I am having some trouble getting my head wrapped around this problem. If anyone can offer suggestions or help it would ...
3
votes
3answers
187 views

Orthogonality relations of Characters

Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following http://www.ms.uky.edu/~pkoester/research/charactersums.pdf Note: The orthogonality relations ...
0
votes
3answers
176 views

Prove abelian group

I am given $((0,1),*)$ Where $x,y\in (0,1)$ and $*$ is defined as $x*y=\frac{xy}{1-x-y+2xy}$ How should I go about finding the inverse and identity elements?
6
votes
3answers
718 views

On the Factor group $\Bbb Q/\Bbb Z$ [duplicate]

Possible Duplicate: $\mathbb{Q}/\mathbb{Z}$ has a unique subgroup of order $n$ for any positive integer $n$? I have the factor group $\Bbb Q/\Bbb Z$, where $\Bbb Q$ is group of rational ...
3
votes
1answer
91 views

Expression of an abelian group

Let $A$ be a abelian group generated by elements $\langle a_1,a_2,a_3\rangle$ and $B$ be a subgroup generated by $\langle b_1,b_2,b_3\rangle$ where $\begin{pmatrix} b_1\\ b_2\\b_3 \end{pmatrix}= ...
0
votes
2answers
240 views

Canonical form of an Abelian group

Given the abelian group : $A=\mathbb{Z}_{36} ×\mathbb{Z}_{96}×\mathbb{Z}_{108}$ I need to write the canonical form of $18A$ and $A / 18A$ Here is my calculation ,using the followings: ...
1
vote
1answer
123 views

Classifying some abelian groups of order $2^5\times 3^5$

I'm requested to classify the abelian groups $A$ of order $2^5 \times 3^5 $ where : $| A/A^4 | = 2^4 $ $ |A/A^3 | = 3^4 $ I need to write down the canonical form of each group . My question is, ...