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# Factor Group

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
8 months, 2 weeks 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. · 8 months ago

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

Unfortunately, false statements cannot be proven in a consistent system. · 8 months ago

Of course they can be! · 8 months ago

How can they? · 8 months ago

I meant they can be proven to be false. · 8 months ago

Yep.

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

Much like this one, which has no proof.

· 8 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. · 8 months ago

Yes. How does that make you feel? · 8 months ago

@Kushal Bose · 8 months ago