Suppose that and are subgroups of a group with . Prove that divides

Let N be a normal subgroup of a group G and let H be a subgroup of G that contains N

i) Show that is a subgroup of

ii) If prove that

I know these should be relatively straightforward and I have some answers but think they're really messy and not sure if what I've done is viable!

For first part using LaGrange I have and then as are both subgroups so so that divides

For i) as all also clearly

to show it has an inverse say such that then (can do this as is normal)

then following same argument for also so

For ii) as we have thus so

If anyone can let me know if what I've put is correct and or whether I've missed crucial steps I'd really appreciate it!

Thanks