# 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
3 years, 6 months 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.

- 3 years, 6 months ago

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

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

- 3 years, 6 months ago

Of course they can be!

- 3 years, 6 months ago

How can they?

- 3 years, 6 months ago

I meant they can be proven to be false.

- 3 years, 6 months ago

Yep.

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

Much like this one, which has no proof.

- 3 years, 6 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.

- 3 years, 6 months ago

Yes. How does that make you feel?

- 3 years, 6 months ago

- 3 years, 6 months ago

@Deeparaj Bhat Check this out

- 3 years, 6 months ago