# Tagged Questions

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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### Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
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### Is there a nice characterization for when the torsion subgroup of a group $G$ is a direct summand?

Pretty much just the title. I'm reading Rotman's Introduction to the Theory of Groups, and he gives an example of an abelian group $G$ such that the torsion subgroup (which he denotes $tG$) is not a ...
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### Equivalent conditions for a group being divisible

I'm being asked to show the following are equivalent conditions of an abelian group $G$: (i) $G$ is divisible (ii) Every nonzero quotient of $G$ is infinite (iii) $G$ has no maximal subgroup I've ...
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### Subgroups of $\mathbb{Z}_p^n$

Is there a nice characterization or construction to list the subgroups of $\mathbb{Z}_p^n$, that is, $\mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$ where $\mathbb{Z}_p$ is the cyclic group of prime ...
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### Q as an additive abelian group has no minimal generating set

$\mathbb Q$ as an additive abelian group has no minimal generating set. I have done this question according to the solution given here. First I took a minimal generating set $S$ of $\mathbb Q$ and ...
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### Can not see the use of Correspondence Theorem

In Hersteins proof of Fundamental Theorem of Finite Abelian Groups, I don´t see the use of Correspondence Theorem. It says that exist some subgroup $Q$ of G such that $T=Q/B$. But Im a bit confused ...
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### Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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### How to proceed in the proof of this statement.

I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear. Let $A$ be a normal ...
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### Commutativity of multiplication of cosets of the commutator subgroup

Take a group $H$ with a non-trivial commutator subgroup, and form the quotient group $H^{ab} = H/H'$. Now, take the cosets of the products of elements $a,b$ and $c,d$: $abH'$ and $cdH'$ in $H^{ab}$. ...
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### Prove a group G is abelian if it satisfies x^2 = x for every x in G

I originally solved this problem by simply noting that x^2 = x implies x=e, so the only element in the group is the identity...but this is wrong. I am now stuck on this idea though and I have tried ...
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### Show that the group is abelian

Let $M$ be a field and $G$ the multiplicative group of matrices of the form $\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$ with $x,y,z\in M$. I have ...
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