# Thread: H and K are subgroups of G, K is also normal in G. I want a counterexample...

1. ## H and K are subgroups of G, K is also normal in G. I want a counterexample...

hello everyone. Can any one help me on this:

H and K are subgroups of G, K is also normal in G.
I want to find a counterexample that HK is not normal in G.

Thanks!!

2. Originally Posted by xxie
hello everyone. Can any one help me on this:

H and K are subgroups of G, K is also normal in G.
I want to find a counterexample that HK is not normal in G.

Thanks!!
$G=S_4, \ H=\{(1) , (1 \ 2) \},$ and $K=V,$ the Klein four group.

3. what is K=V????

4. Originally Posted by xxie

what is K=V????
i mentioned that, didn't i? it's the Klein four group: $V=\{(1), (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3) \}.$

5. oh~thanks! i didn't know what klein group is!haha

6. Originally Posted by xxie
hello everyone. Can any one help me on this:

H and K are subgroups of G, K is also normal in G.
I want to find a counterexample that HK is not normal in G.

Thanks!!
Hi xxie.

Here is a much simpler example. Take $G=S_3,$ $H=\{1,(12)\},$ $K=\{1\}.$

7. Originally Posted by TheAbstractionist
Hi xxie.

Here is a much simpler example. Take $G=S_3,$ $H=\{1,(12)\},$ $K=\{1\}.$
allowing K = (1) will make the problem trivial and non-interesting, isn't it?

8. Originally Posted by NonCommAlg
allowing K = (1) will make the problem trivial and non-interesting, isn't it?
Hi NonCommAlg.

As xxie didn’t state any condition on what type of normal subgroup $K$ should be, my example was a fair one.

But, if you like, how about

$G=D_6=\left<\rho,\sigma:\rho^6=\sigma^2=1,\, \rho\sigma=\sigma\rho^{-1}\right>$

$H=\{1,\sigma\}$

$K=\{1,\rho^3\}=Z\left(D_6\right)$