# Questions tagged [abelian-groups]

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

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### Pure Mathematics questions on Group Theory

I am learning Group Theory but am stuck on this particular practice questions, however I am stuck on a few of these. For the second part I have proved that theta is bijective and that theta(x ** y)=...
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### Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2$ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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### Product $PN$ of normal subgroups is abelian

I am trying to show that every non-abelian group $G$ of order $6$ has a non-normal subgroup of order $2$ using Sylow theory. First, Sylow's Theorem says the number of Sylow $2$-subgroups $n_2$ is ...
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### Prove: If gcd(ord(x),ord(y))=1, then ord(xy)=ord(x) ord(y) when x,y in G (abelian) [duplicate]

$\newcommand{\ord}{\operatorname{ord}}$Let G be an abelian group and let $x,y \in G$ be two elements with finite order. Then, prove that if $\gcd(\ord(x),\ord(y))=1$, then $\ord(xy)=\ord(x) \ord(y)$ ...
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### What does the abelianization mean?

Abelianise each of: (a) $\Bbb Q \times S_4$ (b) $D_{12} \times A_4$ (c) $G \times Z_{10}$, where $G$ is the dicyclic group of order 12 and write down the torsion coefficients of the resulting ...
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### Is there a name for this kind of subgroup?

Let $G$ be an abelian group, $H\subset G$ a subgroup such that if $nx\in H$ for $n\in \bf{Z}$ and $x\in G$ then $x\in H$. Is there a name for subgroups of abelian groups $H$ with this property?
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### Describe the types of groups with 10 elements $(G,•)$ group and $|G|=10$. [closed]

I wrote the divisors of $D_{10}= 1,2,5,10$, but I don't know how I can continue.
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### Let G be a Group and $a,b \in G$. If $a$ and $b$ commute and $|a|$ and $|b|$ are finite. What can be possible values for $|ab|$? [duplicate]

I want to find the order of $ab$, which I have tried to find as follows. Let order of $a$ be $|a|=m$ and that of $b=|b|=n$ Let $x=\text{lcm} (m,n)$, then clearly $\exists$ integers $s , t$ such ...
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### Algebra topic suggestions- Final project [closed]

Good day. My abstract algebra teacher asked me to do a mini project, I'm looking for a topic that covers all (or most) of the following topics: Free modules. Matrix bases and finitely generated ...
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### If $G$ is an Abelian Group of rank $r$ then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r$

So I'm trying to prove that if $G$ is an Abelian Group of rank $r$ (As $\mathbb{Z}$-module) then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r.$ Using the results I know about ...
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### G is a commutative group of order 2*N, how to prove G has a N-order quotient group? [closed]

$G$ is a commutative group of order $2N$, how to prove $G$ has a $N$-order quotient group? Now I know $G$ must has two subgroup, one is order $N$ and another is order $2$; now what should do to prove ...
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### Let $G$ be a non-nilpotent group where all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.

Theorem : Let $G$ be a non-nilpotent group such that all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center. Proof. Suppose that $Z(G)$ is non-cyclic. since $G$ is non-...
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### The group of non zero real or imaginary numbers

I have recently come across with the multiplicative group formed by the union of the real and imaginary axes minus the origin of the complex plane. Surprisingly (at least to me) I couldn't find any ...
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### Show $G\cong \ker(f) \times \mathbb{Z}$ for abelian $G$

First of all, I am aware of the First Isomorphism Theorem but I am not sure how to use it/if it is useful here $G$ is an abelian group and $f:G\rightarrow\mathbb{Z}$ is a surjective group ...
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### Homology with Coefficients of an abelian group

I am trying to see why this is true, this is done in Brown's Cohomology of groups: (In terms of notation $\tau(y)$ is the divided polinomyal algebra). Consider the study of $H_*(G,\mathbb{Z}_p)$, ...
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### If $N$ is a normal subgroup of $G$, and $N \cap [G,G]=\{e\}$, then $N$ is contained in $Z(G)$.

I found an answer to this here: If the intersection of a normal subgroup and the derived group is $\{e\}$, show that $N$ is a subset of $Z(G)$.. However I don't really understand some of the answers ...
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### A $p$-group of exponent $p$

I saw a theorem For odd $p$, a $p$-group possesses a characteristic subgroup $D$ of class at most $2$ and of exponent $p$ such that every nontrivial $p’$-automorphism of $G$ induces a nontrivial ...
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### Explicitly describing the subgroups of $\mathbb{Z}^{3}$
I am interested in understanding all the subgroups of $\mathbb{Z}^{3}:=\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. $\mathbb{Z}^{3}$ a free abelian group of rank three, so all subgroups are free ...
### Prove if there is a group whose order is $p^2$,and it is non-abelian, then it is cyclic [closed]
Suppose the non-abelian group G whose order is $p^2$ where p is a prime number, prove it is a cyclic group. My work: there is a $\tau\not=e$ in the group and the order of $\tau$ is either $p$ or \$p^...