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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Showing that the product group of $G$ and $H$ satisfies the universal property for coproducts in the category of abelian groups $\mathbf{Ab}$

I'm working on another problem of Aluffi's Algebra. Given the category $\mathbf{Ab}$ of abelian groups, the problem is to show that for any two groups $G$ and $H$ the product group $G\times H$ ...
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0answers
34 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
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1answer
35 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
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17 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
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1answer
47 views

Isomorphisms based on Conjugacy Classes

For what groups of the same order are not isomorphic and contain the same conjugacy class? I as well have a more detailed question: For which of those groups are not abelian. The only example I know ...
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41 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 ...
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1answer
24 views

Isomorphisms of LCA Groups

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 ...
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40 views

Finding the group structure of a finite ring

Trying to construct an example I built up this finite ring: $$B=\mathbb{Z}/9\mathbb{Z}[x,y,z,w_1,w_2]/(x^3-1,y^3-1,(x-1)(z+3w_1),(y-1)(z+3w_2),w_1^2,w_2^2,z^2)$$ I need to know the structure of ...
3
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2answers
37 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 .
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31 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 ...
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1answer
15 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 ...
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1answer
45 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. ...
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1answer
51 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 ...
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1answer
70 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, ...
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0answers
47 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
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1answer
41 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 ...
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2answers
93 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 ...
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2answers
32 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 ...
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2answers
40 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 ...
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1answer
29 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 ...
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1answer
13 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 ...
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2answers
112 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 ...
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35 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 ...
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2answers
35 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 ...
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1answer
34 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 ...
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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 ...
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1answer
81 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.
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1answer
56 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 ...
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1answer
51 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 ...
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38 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, ...
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1answer
32 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
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1answer
31 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= ...
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1answer
54 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 ...
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1answer
39 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
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1answer
33 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 ...
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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
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1answer
54 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 ...
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0answers
48 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
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2answers
33 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
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1answer
60 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 ...
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1answer
35 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
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1answer
42 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
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2answers
61 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 ...
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3answers
25 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.
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1answer
45 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 ...
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51 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 ...
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84 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 ...
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11 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
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3answers
132 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
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1answer
44 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 $(\mathbb Z , *)$ ...