Factor Group

Can anyone help me in this question :-

Prove or disprove: If HH is a normal subgroup of GG such that HH and G/HG/H are abelian, then GG is abelian.

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

Note by Kushal Bose
3 years ago

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1 vote

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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=S3G=S_3 and H=A3H=A_3 we see that the hypothesis is satisfied though GG isn't abelian.

Deeparaj Bhat - 3 years ago

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

Unfortunately, false statements cannot be proven in a consistent system.

Agnishom Chattopadhyay Staff - 3 years ago

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Of course they can be!

Deeparaj Bhat - 3 years ago

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@Deeparaj Bhat How can they?

Agnishom Chattopadhyay Staff - 3 years ago

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@Agnishom Chattopadhyay I meant they can be proven to be false.

Deeparaj Bhat - 3 years 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 Staff - 3 years 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 - 3 years ago

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@Deeparaj Bhat Yes. How does that make you feel?

Agnishom Chattopadhyay Staff - 3 years ago

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@Deeparaj Bhat Check this out

Agnishom Chattopadhyay Staff - 3 years ago

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