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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Characterization of finitely generated $\mathbb{Z}$-modules with the property that each submodule is a direct summand

I want to characterize all finitely generated $\mathbb{Z}$-modules $M$ with the property that each submodule of $M$ is a direct summand of $M$. I think the module has to be torsion but I couldn't say ...
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3answers
56 views

An example of a group that that has an order of M that is abelian?

Theory: If G is a finite abelian group, p is prime and p divides the order of G then G has an element of order p. Can anyone think of a counter example for a number n that is not prime, divides the ...
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1answer
27 views

Uniqueness of abelian group structure on a given set and recursive algorithms

If we have some function $f$ under $\mathbb{Z}$ and $$f(a, f(b, c)) = f(f(a, b), c)$$ $$f(a, b) = f(b, a)$$ $$f(a, 0) = a$$ $$f(a, -a) = 0$$ meaning $f$ is an abelian group with an identity element of ...
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1answer
50 views

When is the quotient of two lattices in ${\mathbb Z}^2$ cyclic?

In this question, by a lattice I mean a full-rank subgroup of the group ${\mathbb Z}^2$. What I would like to know is: Can one give a comprehensible description of those lattices ...
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1answer
46 views

Assuming the axiom of choice ,how to prove that every uncountable abelian group must have an uncountable proper subgroup?

Assuming the axiom of choice , how to prove that every uncountable abelian group must have an uncountable proper subgroup ? Related to Does there exist any uncountable group , every proper subgroup ...
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1answer
55 views

Characteristic subgroup of an Abelian-by-Finite Group

Let $G$ be a group such that $A$ is a normal Abelian subgroup and $G/A$ is finite. Is always possible to find an Abelian characteristic subgroup $B$ such that $G/B$ is finite too? Factoring by $G^n$ ...
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1answer
32 views

Does an injection of finitely generated abelian groups always induce a surjection via $Hom(-,U(1))$?

I was recently interested in the following conjecture, which at first sight seemed pretty elementary. Conjecture: Let $i: A \hookrightarrow B$ be an injection into a finitely generated abelian group. ...
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0answers
68 views

Why do mathematicians study elementary abelian groups?

I took two algebra courses that I liked as an undergraduate mathematics major in college, but we never covered elementary abelian groups. I recently got interested in the properties of a group I ...
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2answers
178 views

A finite abelian group has order $p^n$, where $p$ is prime, if and only if the order of every element of $G$ is a power of $p$

Suppose that G is a finite Abelian group. Prove that G has order $p^n$, where p is prime, if and only if the order of every element of G is a power of p. I tried the following route, but got stuck. ...
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1answer
60 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, ...
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1answer
53 views

For a subgroup $H$ of a finite group $G$ , when does $|Aut(H)|$ divides $|Aut(G)|$?

Let $H$ be a subgroup of a finite group $G$, then is it true that $|Aut(H)|$ divides $|Aut(G)|$? What if we also assume $G$ is abelian? (I know that $|Aut(H)| \space \big| \space |Aut(G)|$ if $G$ is ...
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0answers
24 views

Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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2answers
69 views

Group Theory: Showing that a subgroup is isomorphic to a product of groups

I have the following question, where the topic being tested is cosets, order and Lagrange's theorem: Suppose that every element $x$ in a group $G$ satisfies $x^2 = e$. Prove that $G $is abelian. ...
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1answer
53 views

How to prove that a certain set is a coset of a subgroup of a group of characters mod $m$?

I have the following question: Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ ...
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2answers
31 views

Showing that the group is abelian

Let $\sigma = (123456)$ in $S_6$. And let $G = \{e, \sigma, \sigma^2, \sigma^3, \sigma^4, \sigma^5\}$ be a group under operation from $S_6$. Is $G$ abelian? Workings: A group is abelian if it is ...
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1answer
36 views

Why is the group abelian?

Lets say a group $G$ consists of 3 Sylow groups, $H_1,H_2,H_3$. Each of order $p_1,p_2,p_3$, that are prime numbers and different. Since we only have one of each Sylow group for each p, the second ...
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1answer
128 views

Are there any non-obvious colimits of finite abelian groups?

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up ...
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2answers
75 views

Let G be a group, where $(ab)^3=a^3b^3$ and $ (ab)^5=a^5b^5$. Prove that G is an abelian group. Want to specify.

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group? P.S Why cannot not we just cancel ab out of the middle of these expressions? Why can we only ...
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4answers
117 views

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group?

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G$ is an Abelian group. I know that the answer for this question has been already posted and I have seen it. However, could ...
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1answer
99 views

When the endomorphism ring of an abelian group is generated by automorphisms?

Given an abelian group $M$. First I'd like to know if $\text{End}(M)$ is generated by $\text{Aut}(M)$ (as ring, or equivalently, as additive group). Second I'd like to know if it doesn't hold ...
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1answer
43 views

How to classify all the abelian groups with finite exponent?

Let $A$ be an abelian group, the exponent exp$A$ is the least natural number $n$ (if exists) such that $nA=0$ or $+\infty$. The question can be reduced to the case exp$A=p^n$ for a certain prime ...
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1answer
93 views

Intuition for an abelian fundamental group

Any topological group has an abelian fundamental group by the Eckmann-Hilton argument. Is there some intuition behind the fundamental group being abelian that would enable one to predict this ...
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1answer
45 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
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1answer
47 views

List all abelian groups that have order 81 and contain an element of order 27

List all abelian groups that have order 81 and contain an element of order 27. For each, give the primary decomposition and a specific element having order 27. I know $81 = 3^{4}$ so the abelian ...
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1answer
62 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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2answers
34 views

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian.

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I know that $S_{4}$ is not abelian and therefore $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I'm not sure how to show this ...
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2answers
173 views

On subgroups of abelian groups

Let $G$ be a product of $n$ finite cyclic groups. Is every subgroup of $G$ also a product of (at most) $n$ finite cyclic groups ? I do not know the answer to this question, but I'm tempted to say ...
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1answer
43 views

Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
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1answer
46 views

If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
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1answer
22 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
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1answer
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Equivalence of definitions of injective modules

Wikipedia article gives a number of definitions of injective modules, namely: If $Q$ is a submodule of some other left $R$-module $M$, then there exists another submodule $K$ of $M$ such that $M$ is ...
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1answer
38 views

Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
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2answers
121 views

Example of an infinite abelian group having a non-cyclic finite subgroup [closed]

Give example (if exists) of an infinite abelian group having a non-cyclic finite subgroup . Please help
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1answer
191 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
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2answers
33 views

Abelian group, inverse element

What would be the inverse element for this abelian group $[1,2,3,4,...,p-1]$ , in which p is a prime number, with this operation $(a*b)mod.p$? For all $a$ and $b$ of the set. I know the inverse ...
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1answer
52 views

Center of an abelian group

Prove if $G$ is non abelian group, then exists an abelian subgroup $H$ which contains $Z(G)$ and $H≠Z(G)$.
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1answer
22 views

Inverse element of the abelian group (P(M),symmetric difference)

What would be the inverse element in this abelian group: $(P(M),\triangle)$? I know the neutral element is the empty set and I thought the inverse element would be $A^{c}$ for every $A$. Turns out ...
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1answer
35 views

Why homomorph and not isomorph?

Why are the groups $\mathbb{R},+$ and $\mathbb{R}_0^+,*$ homomorph, their mapping function being $ f: x \rightarrow e^x $? Why is this not an isomorphism?
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1answer
47 views

finetely generated Abelien -by-nilpotent group

Let G finetely generated Abelien -by-nilpotent group (i.e there existe a abelien subgroup H in G and G/H is nilpotent )With each of its two-generator is nilpotent-by-finite show that G is ...
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1answer
69 views

Show there is no non-abelian group of order 9 [duplicate]

I want to show there is no non-abelian group of order 9. How should I attempt this?
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1answer
121 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
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2answers
63 views

Non abelian subgroup of a abelian group.

What is the relationship between abelian subgroup of a non-abelian group(when exist, example, theorem)?? any thing such link regarding the question would help. ...
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2answers
28 views

Order and Least Common Multiple Abelian Question

\item Let $G$ be an abelian group and let $x, y\in G$ be elements so that $o(x)=m$ and $o(y)=n$. Show that $o(xy)=\frac{mn}{(m,n)}$. (Note that this is the least common multiple of $m$ and $n$) Is ...
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1answer
66 views

How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is ...
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1answer
30 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
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1answer
29 views

$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
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5answers
72 views

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? [duplicate]

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? Since $(ab)^n=e$, I get $(b)(ab)^n(a)= ba$. This means that $(ba)^{n+1}=ba$, and $(ba)^n=e$. But, I ...
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0answers
47 views

Let $G$ be a group and let $\Phi\colon G \to G$ be an isomorphism. Define $H = \{ a \in G\ |\ \Phi(a) = a^{-1} \}$ [duplicate]

It asks to prove that if $H$ is a subgroup of $G$, then $G$ is abelian. Solution: I showed that for every $a$ and $b$ in $H$, $a$ and $b$ commute. But how do I generalize to elements in $G$ NOT in ...
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2answers
54 views

Prove rk$B$ $\le$ rk$A$ where A and B are free, abelian and finitely generated groups.

Let $A$ and $B$ be free abelian, finitely generated groups. Let $f:A \to B$ be an epimorphism. Prove rk$B$ $\le$ rk$A$. I could really use a verification. That is a question from my exam today. ...
2
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0answers
61 views

$gcd(|G|, |Aut(G)|)=1$ means G is abelian?

Prove the following assuming that G is finite group with $gcd(|G|, |Aut(G)|)=1$ a)G is abelian (done) b) Every Sylow subgroup of G is cyclic of prime order. G is abelian than every sylow unique, ...