# 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
4 years, 1 month 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.

- 4 years ago

- 4 years ago

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

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

Of course they can be!

- 4 years ago

How can they?

I meant they can be proven to be false.

- 4 years ago

Yep.

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

Much like this one, which has no proof.

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

- 4 years ago

Yes. How does that make you feel?

@Deeparaj Bhat Check this out

- 4 years, 1 month ago