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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.

Note by Kushal Bose
1 year ago

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

Well, 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 · 12 months ago

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@Deeparaj Bhat

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

Unfortunately, false statements cannot be proven in a consistent system. Agnishom Chattopadhyay · 12 months ago

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@Agnishom Chattopadhyay Of course they can be! Deeparaj Bhat · 12 months ago

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@Deeparaj Bhat How can they? Agnishom Chattopadhyay · 12 months ago

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@Agnishom Chattopadhyay I meant they can be proven to be false. Deeparaj Bhat · 12 months ago

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@Deeparaj Bhat Yep.

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

Much like this one, which has no proof.

Agnishom Chattopadhyay · 12 months ago

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@Agnishom Chattopadhyay Yes. In any consistent system there exist statements which can't be proved or disproved. To disprove here means to prove the negation. Deeparaj Bhat · 12 months ago

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@Deeparaj Bhat Yes. How does that make you feel? Agnishom Chattopadhyay · 12 months ago

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@Deeparaj Bhat Check this out Agnishom Chattopadhyay · 1 year ago

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