Thread: Frobenius Groups

I have searched through 2 books by Ledermann on group theory and cannot find any proofs to these questions.
Can anybody help?

1.Prove that a finite Group G with a nilpotent subgroup of prime-power
index pn say, is soluble.
2. Show that the centre of a Frobenius group is always trivial.

Originally Posted by Turloughmack

I have searched through 2 books by Ledermann on group theory and cannot find any proofs to these questions.
Can anybody help?

1.Prove that a finite Group G with a nilpotent subgroup of prime-power
index pn say, is soluble.
2. Show that the centre of a Frobenius group is always trivial.

For 1. do you have normality? If so, so G/H is nilpotent, and so is H, so...

For 2. I'll give you a hint: how does the centre act on H by conjugation, where H is the subgroup such that ?