Thread: unique subgroups

Prove that has a unique subgroup of order 4.

I know all 12 elements of , and I know contains , but how do I prove that is a unique subgroup?

Originally Posted by dori1123

Prove that has a unique subgroup of order 4.

I know all 12 elements of , and I know contains , but how do I prove that is a unique subgroup?

Since subgroups of order are Sylow -subgroups it follows all subgroups of order must be conjugate by Sylow's second theorem. Now is a normal subgroup of . Therefore, it must be the only one.