Thread: Group of order 30

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.

Originally Posted by jackie

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.

let be two subgroups of with you've proved that either or is normal. thus is a subgroup of clearly and so

now recall that every subgroup of index 2 in any group is normal.