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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?

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  • $\begingroup$ For your first quote, Thompson says Higman recently gave a proof and gives a reference. Did you check that reference? $\endgroup$
    – verret
    Commented Aug 5 at 2:49
  • $\begingroup$ 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. $\endgroup$
    – verret
    Commented Aug 5 at 2:52
  • $\begingroup$ @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 $\endgroup$
    – Quay Chern
    Commented Aug 5 at 3:20
  • $\begingroup$ 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! $\endgroup$
    – verret
    Commented Aug 5 at 3:55
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    $\begingroup$ 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). $\endgroup$
    – Steve D
    Commented Aug 5 at 11:26

1 Answer 1

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For the first result, Thompson says the proof can be found in Higman. He gets the name of the paper slightly wrong, but otherwise the reference is fine, the result is Theorem 4 in the paper of Higman.

As for Theorem A, it's the main result of "Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift, 72: 332–354".

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