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

learn more… | top users | synonyms

1
vote
2answers
46 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
1
vote
1answer
47 views

Error in Cariolaro's Unified Signal Theory

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
5
votes
2answers
69 views

To prove , if Aut$ (G)$ is trivial then $x^2=e , \forall x \in G$

If for a group $G$ the only automorphism is the identity automorphism , then how do we prove that $x^2=e ,\forall x \in G $ ? I have only been able to prove that $G$ is abelian ; Please Help .
0
votes
0answers
43 views

group of order $36$ abelian

Let $G $ be a group of order $36$.How to conclude whether it is abelian or not .I tried using Sylow's theorems by calculating the number of subgroups of order $4$ and $9$ but I am getting so many ...
1
vote
1answer
75 views

Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
0
votes
1answer
48 views

If Abelian Group $G \cong G^3$, does it follow $G \cong G^2$?

If an Abelian Group $G$ satisfies $G \cong G^{3}$, does it follow that $G \cong G^2$ ? It seems elementary but I can't find it on a standard textbook exercise, and maybe simply because it's false. ...
6
votes
1answer
57 views

If N and every subgroup of N is normal in G then G/N is abelian .

Let $N$ be a normal subgroup of $G$ such that every subgroup of $N$ is normal in $G$ and $C_G(N)\subseteq N $ .Prove that $G/N$ is abelian. I think we need to use that every subgroup of $N$ is ...
1
vote
1answer
77 views

Why is $G$ abelian?

If $|G|=pq^2$ with $p,q$ primes and if $p<q$, with $q\not\equiv\pm1\mod p$, why is $G$ abelian ? The $3^{rd}$ Sylow theorem implies that $n_p|q^2$ and $n_p\equiv 1 \mod p$, By hypothesis, ...
3
votes
0answers
54 views

Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and ...
4
votes
1answer
60 views

Finite abelian groups and subgroups.

Let $G$ be a finite abelian group of order $n=p_1^{a_1}\cdot \cdot \cdot p_k^{a_k}$ and $H$ a subgroup of $G$ of order $m=p_1^{b_1}\cdot \cdot \cdot p_k^{b_k}$. By Theorem 5 on page 161 of Dummit ...
2
votes
2answers
112 views

Prove that $G$ has a subgroup isomorphic to $G/H$.

Let $G$ be a finite abelian group of order $n$ and let $H$ be a subgroup of $G$ of order $m$. Show that $G$ has a subgroup isomorphic to $G/H$. Here are my thoughts: Define $\mu_n := \{z \in ...
1
vote
2answers
37 views

Show G is abelian iff the componentwise product homomorphism condition is satisfied:

Let $(G, \cdot, e)$ be a group. For homomorphisms $\theta: \mathbb{Z} \rightarrow G$, $\psi: \mathbb{Z} \rightarrow G$, define the componentwise product as $\theta \cdot \psi: \mathbb{Z} \rightarrow ...
1
vote
2answers
49 views

Subgroup of finite abelian group of order m

I am trying to prove the below statement: Let G be an abelian group of order m. If n divides m, show that G has a subgroup of order n. I think the classification theorem for finite abelian groups ...
0
votes
1answer
42 views

Characters of Finite Abelian Groups

I am studying this proof in my algebra notes, and I would like some help regarding the requirements of the proof. The statement of the proof is: For each finite abelian group G and each h in G with ...
0
votes
1answer
16 views

trying to find associativity

Is the binary operation define by: $x*Y = x+y-1$ what my tutor has done: $x*(y*z) = x *(y+z -1) = x+(y+z-1) = x+y+z-2$ My question: how did he get $x+y+z$-2 Where did the '-2' come from? I am ...
4
votes
2answers
121 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
0answers
45 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
92 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
40 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
84 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
103 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
67 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
65 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
51 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
82 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
35 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
66 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
52 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
vote
1answer
37 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
40 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
62 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
53 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
38 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
112 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
41 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
2
votes
1answer
48 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
165 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
29 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
62 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
73 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
91 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
28 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
165 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
49 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
31 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
65 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
68 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
35 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
123 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 ...