Question

Suppose that H and K are subgroups of G. Prove that if both H and K are normal subgroups of G, then HK is a normal subgroup.

Answer 1

So from your previous problem we know that if at least one of H or K is a normal subgroup of G, then HK was a subgroup of G.

Here, we're now working with the assumption that both H and K are normal subgroups of G. What you need to show now is that for any \(\large g\in G\) and any \(\large hk\in HK\), \(\large g^{−1}(hk)g \in HK\). Then we can conclude that \(\large HK \trianglelefteq G\). Can you see how to use the normality of H and K here to get the result? (The proof can be done in one or two lines.)

I hope this makes sense!

Answer 2

would it be the same as using the normality in the other question?

Answer 3

It should be similar; in fact, it's applied more easily here than in the other problem.

Answer 4

thank you so much. I will look back over my book example to make sure I work it right