Can anyone help me in this question :-

**Prove or disprove**: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are
abelian, then \(G\) is abelian.

I don't need any counter example. I need a proper proof.

Can anyone help me in this question :-

**Prove or disprove**: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are
abelian, then \(G\) is abelian.

I don't need any counter example. I need a proper proof.

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## Comments

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TopNewestWell, it turns out that giving a counter example is a

proper proof.Taking \(G=S_3\) and \(H=A_3\) we see that the hypothesis is satisfied though \(G\) isn't abelian. – Deeparaj Bhat · 10 months ago

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Unfortunately, false statements cannot be proven in a consistent system. – Agnishom Chattopadhyay · 10 months ago

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– Deeparaj Bhat · 10 months ago

Of course they can be!Log in to reply

– Agnishom Chattopadhyay · 10 months ago

How can they?Log in to reply

– Deeparaj Bhat · 10 months ago

I meant they can be proven to be false.Log in to reply

But I suppose there are statements which cannot be proven or disproven.

– Agnishom Chattopadhyay · 10 months agoLog in to reply

– Deeparaj Bhat · 10 months ago

Yes. In any consistent system there exist statements which can't be proved or disproved. To disprove here means to prove the negation.Log in to reply

– Agnishom Chattopadhyay · 10 months ago

Yes. How does that make you feel?Log in to reply

@Kushal Bose – Deeparaj Bhat · 10 months ago

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@Deeparaj Bhat Check this out – Agnishom Chattopadhyay · 10 months ago

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