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I'm stuck on the following practice problem. Any hints would be appreciated.

Suppose $N$ is a normal subgroup of $G$ such that every subgroup of $N$ is normal in $G$ and $C_{G}(N) \subset N$. Prove that $G/N$ is abelian.

I'm not sure how to use the fact that $C_{G}(N) \subset N$.

Thanks

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    It would be way more useful if you'd posted your insights, ideas, effort, background and/or things you already know about this problem. -12012-09-01

2 Answers 2

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Let $n\in N$, and consider the action of $G$ on $\langle n\rangle$. This embeds $G/C_G(\langle n\rangle)$ into $Aut(\langle n\rangle)$, an abelian group. Doing this for all cyclic subgroups of $N$ gives an embedding of $G/C_G(N)$ into a direct product of abelian groups. We are done then, because that means $G/C_G(N)$ is abelian, and $G/N$ is a quotient of that group.

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    "The action"...you mean, I presume, *the action by conjugation* , right? I guess the OP could know this, but it is not immediate from his post, which gives no background, ideas, etc. at all, and not everybody knows about the injection $$N_G(H)/C_G(H) \to Aut(H)$$2012-09-01
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First of all, don't get stuck on what is given. This is the wrong place to look when you start on a proof. Rather, you should look at what you need to prove. In this case, we want to show that $G/N$ is abelian. What does it mean for a group to be abelian?

Well, the definition states that a group $G$ is abelian if for all $g, h \in G$ we have $gh = hg$. So this means we need to pick any two elements from $G/N$ and show that they commute under the group's operation.

I'll let you think about it from there. Let me just emphasize that whenever you write a proof, you need to start with the definition of what you are trying to prove. This almost always gives you a guide as to how to start your proof.

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    I downvoted, as this is not even close to an answer.2012-09-01
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    @SteveD Sure it isn't a complete answer. It's too lengthy for a comment, though, and it points the OP in the right direction to solve this problem on their own. A "practice problem" sounds like homework, so giving a complete answer would be doing the OP a disservice, IMO.2012-09-01
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    It's not an answer in any way, it just rewords some definitions.2012-09-01
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    @SteveD Isn't that what many proofs are?2012-09-01
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    @Code-Guru Yes, that is what many proofs are. Is that what a proof of *this problem* is, though? Everything you wrote is blatantly obvious to me and in no need of mention, yet I still don't personally see how to prove the result - I imagine this is not just me either. As such, it seems this answer is useless: no one with the necessary group theory background to understand the question will be any closer to finishing the exercise than they were when the were reading the question to begin with.2012-09-01
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    @anon Given what the OP has posted, there isn't much evidence to make any assumptions about their group theory background. It certainly would help if they posted what they have attempted as a proof. I read the question as the OP doesn't even know where to start. Thus pointing them to start with the definition seems like a logical answer to me. It might not lead in the right direction, but it is certainly worth the effort in understanding what one is trying to prove.2012-09-01
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    The OP said they were stuck in using a specific fact, and made no mention of not knowing or understanding basic definitions (why would an OP do the former but not the latter if they did not feel that they know or understand the definitions of the terms being used and what the question is asking for?) - and the fact that centralizers are being discussed at all should signify the OP is at a level where they know what abelian means.2012-09-01
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    I agree with anon: the OP's post history shows he understand what a commutator is, and the info in the OP's profile says they are a grad student, preparing for comps.2012-09-01
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    To rely on reading the OP's post history is too stretching the work one could do to *guess* the OP's background: I agree with Guru in this, the OP **should** have posted his question with way more ideas, background, things already known, etc. Yet, I think Steve's point is the main one here: Guru's writing doesn't come even close to be anything ressembling a hint of a possible answer (he didn't even mentioned that $\,G/N\,$ abelian $\,\Longleftrightarrow G'\leq N\,$...!) , and I'd advice him to delete his post as it is useless and will probably bring upon him lots of downvotes.2012-09-01