# finite groups with fixed-point-free automorphisms of prime order

I'm reading the thesis of John Thompson, finite groups with fixed-point-free automorphisms of prime order. This is the link: finite groups with fixed-point-free automorphisms of prime order.

In his paper, it states that: "The more detailed study of $$N$$ leads to the result that if $$N$$ is solvable, then it is even nilpotent." The $$N$$ here is the Frobenius kernel, a group possesses a fixed point free automorphism of prime order. But I found no reference which could give a proof of this result.

And there is also a "THEOREM A" which has not been proved in the paper: "Let $$G$$ be a finite group with a $$p$$-Sylow subgroup $$P$$, $$p$$ is an odd prime, and let $$\mathfrak{A}$$ be a group of automorphisms of $$G$$ which leaves $$P$$ invariant. Suppose for every $$\mathfrak{A}$$-invariant normal subgroup $$Q$$ of $$P$$, elements of order prime to $$p$$ which normalize $$Q$$ also centralize $$Q$$. Then $$G$$ possesses a normal $$p$$-complement."

Are there any references of the two results above?

• For your first quote, Thompson says Higman recently gave a proof and gives a reference. Did you check that reference? Commented Aug 5 at 2:49
• As for Theorem A, I'm not sure, but according to Wikipedia, Thompson proved that a Frobenius kernel is nilpotent in "Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift, 72: 332–354". That's just one year later than the paper you are reading, so would be a good place to look. Commented Aug 5 at 2:52
• @verret Thanks, I have found the paper of Higman, but the paper's name is not the same as the one in Thompson's reference: Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements Commented Aug 5 at 3:20
• Well, the name is slightly different, but it's clearly the right paper (the rest of the reference is correct). Moreover, the first result is there, as Theorem 4! Commented Aug 5 at 3:55
• If you're looking for a proof of theorem 4, it is Theorem 6.22 in Isaacs's Finite Group Theory (which also proves Thompson's normal p-complement theorem). Commented Aug 5 at 11:26