# Questions tagged [abelian-groups]

For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.

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### Why is a Sylow 5-subgroup abelian?

For weeks I tried to solve the following question on Brilliant: Fill in the blank: "Every group of order ___ is abelian." And these are the possible answers I get: 15, 16, 20, 21, 27. Using ...
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### Let $h\in G$. Let $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$. Then $G$ is abelian iff $\phi_h = \operatorname{id}_G$ for all $h\in G$.

Let $G$ be a group and $h\in G$. Consider the map $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$, $G$ is an abelian group if and only if $\phi_h = \operatorname{id}_G$ for all $h\in G$. I know what it ...
1 vote
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### How to show that a subgroup of a product of finite cyclic group has itself this property?

Let $G$ be a subgroup of a finite product of finite cyclic groups. Is it easy to prove that $G$ is itself a finite product of cyclic groups without appealing to the structure theorem of finite abelian ...
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### On Ulm invariants of a famous abelian p-group due to Prufer

We know that there is a famous example due to Pruefer, of a countable reduced p-group $G$ of length $\omega + 1$ as following:// Let G be the Abelian group generated by the countable set of ...
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### How to find invariant factor of this finitely generated group

I'm trying to solve this problem. Let $G$ be the set $\Bbb Z^2$ with binary operation defined by $(x_1,y_1)*(x_2,y_2) = (x_1+x_2, y_1+y_2+x_1x_2)$ Show that $(G,*)$ is a finitely generated abelian ...
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### Finding all the possible orders of elements in a quotient group

Let $A$ be an arbitrary Abelian group and let $H:=\{a\in A\mid\exists b\in A\mid a=b^3\}.$ Prove $H$ is a normal subgroup of $A$. Determine all the possible orders of elements in the group $A/H.$ My ...
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### The maximal free abelian subgroup that can be embedded in $GL(n,\mathbb{Z})$

I am stuck on this problem and cannot seem to find a good reason for drawing the required conclusion. The problem is as follows: Given $SL(n, \mathbb{Z})$ a subroup in $GL(n, \mathbb{Z})$. How can ...
1 vote
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### A group closed under scalar multiplication is a vector space

Suppose $G$ is an abelian group closed under scalar multiplication with elements in the field $F$. Is $G$ always a vector space over $F$? I have been trying to find a counter-example, but failing. In ...
1 vote
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### Abelian groups with all elements having order $2$

This result is not an exercise, but more so a hunch that I am trying to prove for myself. Let's say that $G$ is a finite group with the property that it is both abelian and every non-identity element ...
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### Doubt regarding the generality of an equivalent definition for Abelian Group [duplicate]

I was given a problem as follows: Let $G$ be a finite group of odd order. If $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b\in G$ then $G$ is abelian. I was able to show that this is ...
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### Proving that there is a functor $F: Grp \rightarrow Ab$ s.t $F(G)=G_{ab}$- How to deal with the quotient?

Let $G$ be a group. $f$ a group hom. $f:G\rightarrow H$ I define: $F(G)=G_{ab}$ and $F$ : $G$ $\rightarrow$ $G_{ab}$ a homomorphism, where $G_{ab} : = G/[G,G]$ is the abelianization of group $G$....
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### When two infinite direct products $\prod_I\Bbb{Z}$ and $\prod_J\Bbb{Z}$ are isomorphic?

It is known that two free $\Bbb{Z}$-modules $\bigoplus_{I}\Bbb{Z}$ and $\bigoplus_{J}\Bbb{Z}$ are isomorphic if and only if $|I|=|J|$. Moreover, it is true for any two free module over a commutative ...
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### Abelian, transitive subgroup of $S_A$, then $\sigma(a)\neq a$ for $\forall\sigma \in G-\{1\}$ and $\forall a$

I have read these questions: Show that $\sigma(a)\ne a,\forall\sigma\in G-\{1\}$ and all $a\in A$.where $G$ is abelian, transitive subgroup of $S_A$, Show that any abelian transitive subgroup of $S_n$ ...
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1 vote
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### What is ${\rm Aut}\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$?

Let $p$ be a prime, what is the automorphism group of $G=\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$ (countable infinite direct sum of $\mathbb{Z}_p$)? I know every permutation of the coordinates will ...
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### $\prod_p \mathbb{Z}/p\mathbb{Z}$ is not the direct sum of $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ and a torsion-free subgroup

While I was reading "Abelian Groups" by Fuchs $(2015)$, I encountered Example $1.2$ in the chapter on Mixed Groups, which stated the following: Let $p_1,p_2,\dots,p_n,\dots$ denote different ...
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### Comparing spectral sequences with different coefficients

Several spectral sequences coming from geometry have a "degree of freedom" in choosing the coefficients (say an abelian group $A$). This applies fo example to the Serre spectral sequence and ...
It's well known that an elliptic curve of the form $y^3 = x^3 +ax +b$ admits a group structure, as long as a point at infinity $O$ is added to serve as identity. If we look at a real elliptic curve as ...
### $\underset{p\in \mathbb{P}}{\prod}\mathbb{Z}/p\mathbb{Z}$ is non splitting mixed abelian group.
We say that an abelian group $G$ is mixed if it has elements $\neq 0$, that are of finite order (torsion elements), as well as elements of infinite order (torsion-free elements). We denote the ...