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
114 views

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$?

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$? Clearly, we can assume the Structure Theorem for finite abelian groups. Edited Later: All I ...
0
votes
1answer
38 views

Show that $a\mapsto a^n$ is an isomorphism when $\gcd(|G|,n)=1$.

Let G be a finite abelian group and let n be a positive integer that is relatively prime to $|G|$. Show that the mapping $\phi:G\to G$ given by $a\mapsto a^n$ is an isomorphism. I solved homomorphism ...
0
votes
2answers
45 views

How many subgroups or order 8 an abelian Group of order 72 can have

Let $G$ be an abelian group of order 72.How many subgroups of order 8 and 4 can have?? I have listed all possible abelian groups there are 6.Then i said that if im lookin for an abelian group of order ...
1
vote
1answer
35 views

Subgroups of Order $p^2$ in $\mathbb{Z}_p \oplus \mathbb{Z}_p$

Hello Mathematics Community. I am unsure about how to solve this problem involving the number of subgroups in an abelian group. How many subgroups of order $p^2$ does the abelian group $\mathbb{Z}_p ...
2
votes
2answers
78 views

Properties possessed by $H , G/H$ but not G

i) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are abelian but $G$ is not ? ii) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are cyclic ...
-3
votes
1answer
82 views

Let G be a group satisfying $a^2 = 1$ for all a in G. Show that G is abelian. [duplicate]

Let $G$ be a group satisfying $a^2 = 1$ for all $a \in G$. Show that $G$ is abelian.
0
votes
1answer
58 views

Any two abelian group of order 8 must be isomorphic

TRUE/FALSE :Any two abelian group of order 8 must be isomorphic SOLUTION: True The problem of finding all abelian groups of order 8 is impossible to solve, because there are infinitely many ...
3
votes
1answer
54 views

On the Definition of multiplication in an abelian group

In class we had the following Definition: Let $(A,+)$ be an abelian Group with $a \in G$. We define: $$na:= \begin{cases}na, \ \forall n \in \mathbb{N} \\ |n|(-a), \ \forall n \in ...
2
votes
0answers
40 views

Classify $\mathbb{Z_6} \times \mathbb{ Z_{24}} / \langle(3,2)\rangle$ according to fundamental theorem of finitely generated abelian group

The order of $G/H = 12$ So it can be isomorphic to $\mathbb{Z_3} \times \mathbb{Z_4}$ or $\mathbb{Z_3} \times \mathbb{Z_2} \times \mathbb{Z_2}$ $(0,1)$ has order of 4, $(1,0)$ has order of 12, ...
0
votes
1answer
40 views

If G is abelian and simple ,then G is cyclic

True /False .IF G is abelian and simple ,then G is cyclic Solution True If G is an abelian simple group then G is isomorphic to Zp for some prime p
0
votes
1answer
33 views

On the construction of an $R$-Module

Let $X \neq \emptyset$ be a set and $(R,+, \cdot)$ a commutative Ring with $\mathbb{1}$ and $(N,+, \cdot)$ an $R$-Module. Show that $(\text{map}(X,N), +, \cdot)$ is an $R$-Module where for $A= ...
1
vote
1answer
56 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
0
votes
1answer
44 views

abelian group as Z module

How Would you prove that every abelian group can be understood as a Z-Module in a unique way? I would guess that you would have to prove its bijective, but not sure how to go about this
1
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1answer
34 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
0
votes
1answer
37 views

Calculate factor group $(\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z})/\langle(1,1,1)\rangle$

My instructor's said at the lecture that you basically set one of the dimensions to $0$ and hence you get $\mathbb{Z} \times \mathbb{Z}$. Again, is there a better way to think about this problem and ...
2
votes
1answer
60 views

Let $R$ be a $M\times N$ matrix with rational entries. Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under $R$. Consider an equivalence relation on $R\mathbb{Z}^N$ defined by $a\sim b$ if $a-b\in ...
0
votes
0answers
50 views

Non finitely-generated projective $\mathbb{Z}$-module [duplicate]

Let $M$ be a projective $\mathbb{Z}$-module. Must $M$ be free? It is easy to see that the answer is yes if $M$ is finitely generated, but I do not know about the general case. If the answer ...
2
votes
2answers
34 views

Compatibility of homomorphisms and quotient maps of abelian groups

Suppose $A$ and $C$ are abelian groups with subgroups $A'$ and $C'$ respectively. Let $f:A\to C$ be a group homomorphism. I was wondering if the following statements are equivalent: There exists a ...
2
votes
1answer
68 views

Subgroup of finitely generated abelian group is finitely generated

Call a group $G$ finitely generated if there is a finite subset $X \subseteq G$ with $G = \langle X \rangle$. Prove that every subgroup $S$ of a finitely generated abelian group $G$ is itself finitely ...
1
vote
1answer
38 views

Free Group Norms

Hello everyone, I'm trying to solve this problem, but I'm stuck... i don't quite understand the definition of the norm, If you guys can give me a better explanation, I would appreciate it, Thanks
1
vote
1answer
44 views

A question about cyclic Abelian group

If $G$ is a finite Abelian group and for any prime $p$ divides $|G|$ there exists exactly one subgroup of order $p$ in $G$. Suppose $G_p=\{x\in G|x \text{ is a p-element}\}$, then prove $G_p$ is ...
6
votes
2answers
71 views

Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
0
votes
3answers
28 views

Isomorphisms in finite abelian groups 1

True of false? If G and H are two groups with the same order and both are abelian, then they are isomorphic.
0
votes
1answer
50 views

All Isomorphic Classes of Abelian Groups of Order $n$

I know that each finite abelian group is isomorphic to a direct product of cyclic groups of prime orders $> 1$. This means taking a finite abelian group of order $n$, I can find the prime ...
1
vote
0answers
56 views

Verifying proof that set of all group homomorphisms is an abelian group

I'm working on a proof to show that for a group $G$ and an abelian group $H$, the set of all homomorphisms $\def\Hom{\operatorname{Hom}}\Hom(G,H)$ from $G$ to $H$ is an abelian group. I just want to ...
1
vote
0answers
85 views

Prove that module has finitely many elements

Let $p$ be a prime number. Consider the subring $U:= \mathbb{Z}[1/p]$ of $\mathbb{Q}$ and define the $\mathbb{Z}$-module $M:=U/ \mathbb{Z}$ (1): Show that any $\mathbb{Z}$-submodule of $M$ that is ...
1
vote
0answers
12 views

Orthogonality Relations for Character Groups

I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link: ...
2
votes
3answers
138 views

Identity and Inverse Homomorphisms

For a group G and an abelian group H, Hom(G,H) is the set of all homomorphisms from G to H. My notes from class talk about the identity and the inverse homomorphism- I was wondering what these are? ...
2
votes
1answer
45 views

Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
0
votes
1answer
26 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
2
votes
2answers
55 views

Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
0
votes
1answer
57 views

Free group on two generators and commutators. Why it's enough to add the relation ab=ba?

I've looked through lots of question on this topics, but I cannot find what I want to prove: I've seen in a lots of exercises sheets that the abelianization of a free group with two generators (let's ...
3
votes
0answers
76 views

A group is abelian [duplicate]

Let $G$ be a finite group. For any two elements $a,b \neq e\in G$ there exits an automorphism $\sigma$ such that $\sigma(a)=b$. Prove that $G$ is abelian. Only thing that I could conclude about the ...
0
votes
1answer
29 views

Finite Abelian Group Proof

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for some prime $p$. I'm not sure what to do. Any proofs or hints ...
3
votes
3answers
113 views

Is every free subgroup a direct summand?

Let $G$ be an abelian group. Suppose that $F$ is a subgroup of $G$ such that $F$ is free. Does there necessarily exist a subgroup $H\subset G$ such that $G\cong F\oplus H$? Motivation: In Lang's ...
2
votes
1answer
70 views

homomorphisms of abelian groups

Describe: 1) Hom(Q/Z->Q) 2) Hom(Q->Q/Z) Q rational numbers Z integers My thoughts 1) Q/Z i can describe as a/b, a,b coprime and smaller than 1. So I thought Hom(Q/Z->Q) I can describe as [f:a+N->b, N ...
0
votes
4answers
31 views

Automorphism iff G is abelian

Let $G$ be a group. Prove the mapping $\alpha(g)=g^{-1}\forall g \in G$ is an automorphism iff $G$ is abelian. Proof (forwards): Assume $G$ is an automorphism. Show $ab=ba$. How would I even go about ...
2
votes
3answers
153 views

How many distinct subgroups of order 10 are there in a non-cyclic abelian group of order 20?

We are currently working with free abelian groups and finitely generated groups. The homework problem asks us to find the number of distinct subgroups of order 10 in a non-cyclic abelian group of ...
1
vote
0answers
33 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
1
vote
2answers
100 views

Order of the group.

An abelian group $G$ is generated by $x$ and $y$ with $$O(x)=16,O(y)=24,x^2=y^3$$ What is the order of $G$? My attempt:There are $24+16-1=39$ elements generated by $x$ and $y$ separately. Also ...
1
vote
0answers
72 views

Description of an abelian group

I'm again stuck in an algebra exercise. I'm not sure if I understand the problem right. Could it be that I have to show that $\mathbb{Z}[i]/\gamma$ can be expressed by a product of finite abelian ...
1
vote
1answer
31 views

Can we calculate the order of $\hom (G,G')$ in terms of $|G| $ and $ |G'|$ , when $G,G'$ are finite abelian groups?

Let $G,G'$ be abelian groups and let $\hom (G,G')$ be the set of all homomorphisms from $G$ to $G'$. We define an operation $\ast$ on $\hom (G,G')$ as: for $f,g \in \hom(G,G') \space , (f\ast ...
0
votes
2answers
37 views

|ab|=lcm(|a|,|b|) in an abelian group

Assume in an abelian group $G$ that $\langle b\rangle\cap \langle a\rangle=e$, then the order of $(ab)$ is the lcm of the orders of $a$ and $b$. Essentially, $|ab|=\operatorname{lcm}(|a|,|b|)$. So ...
3
votes
2answers
89 views

Irreducible subgroups of the additive rationals

Let $G$ be a group. A proper subgroup $H$ is called irreducible if $H$ can't be written as an intersection of two subgroups which contain it properly. I'd like to know if $(\mathbb Q,+)$ (and ...
1
vote
2answers
44 views

Least common multiple of orders and abelian groups.

I am a little stuck here and would like some minor help. The quesiton I am dealing with is: Assume in an abelian group G that $<b>{\large\cap} <a>=e$, then the order of $(ab)$ is the lcm ...
0
votes
1answer
92 views

$G$ infinite abelian group with $[G:H]$ finite for every non trivial subgroup $H$ , to prove $G$ is cyclic

Let $G$ be an infinite abelian group such that for any non-trivial subgroup $H$ of $G$ , $[G:H ]$ is finite ; then how to prove that $G$ is cyclic ? Please don't use any structure theorem of abelian ...
5
votes
1answer
229 views

Direct sum and direct product of infinitely many abelian groups are not isomorphic

Let $I$ be an infinite set, and for each $i$ let $A_i$ be an abelian group with order $o(A_i) \ge 2$. Prove that the direct product $\prod A_i$ and the direct sum (coproduct) $\bigoplus A_i$ are ...
1
vote
1answer
27 views

Show $a\in G$ is contained in $Z(G)$ iff $Z(a)=G$ for center and centralizer? [closed]

The center of a group $G$ is defined as the set $Z(G):= \{a\in G\mid \forall b\in G : ab=ba\}$ and the centralizer of an element $a\in G$ is defined as the set $Z(a) := \{b\in G\mid ab=ba\}$. How can ...
0
votes
1answer
24 views

The Dihedral group $D_1$ is non-abelian?

Same as above. I'm trying to show that for any n being odd, $D_n$ has exactly n elements of order 2 where $D_n$ is non-abelian. I know that for $n\ge3$ this is true, but what about for $n=1$.
-2
votes
2answers
73 views

Two abelian groups with the same order are isomorphic? [closed]

True of false: if G and H are two groups with the same order and both are abelian, then they are isomorphic.