I am working on showing that there are exactly 4 non-isomorphic groups of order 30. I'm stuck on showing that this group of order 30,
must have a normal subgroup,
of order 15.
I have
Then by using Sylow-theorems I have the possibilities for Sylow p-subgroups of G as followed:
Then I use counting argument to show that it can not be the case that
and
. So, either
or
. Then it follows that
has either a normal Sylow 3-subgroup or a normal Sylow 5-subgroup. But I can't argue that it must have both of those.
I notice that by Cauchy, G must also have a subgroup of order 2, but didn't see how this helps.
I really appreciate any help.