You don't need any of that.
and are all you need.
I want to use the "one step subgroup test".
The identity is
And
and
Therefore H is not empty since .
Now I have to show that for any two elements , is in H.
Let
So we have:
And
If my working is correct so far, could anyone please show me how to manipulate this determinant to show that it satisfies the given condition, and .