From: mareg@lily.csv.warwick.ac.uk (Dr D F Holt)
Subject: Re: Sylow subgroup question
Date: 22 Feb 1999 10:42:11 GMT
Newsgroups: sci.math
Keywords: Fixed-point-free automorphisms of groups
In article <01be5e40$24331d00$37b939c2@buromath.ups-albi.fr>,
"pascal ORTIZ" writes:
>I'm stuck on the following entomological type question
>on Sylow subgroup :
What do you mean by 'entomological'? According to the OED, the only
meaning is 'pertaining to the study of insects'.
>Let G be a finite group and let f:G-->G a group automorphism
>such that f^3=Id and ( f(x)=x ==> x=1).
>Prove G has a unique Sylow p-subgroup for each prime p.
This is proved in Theorem 1.5, Chapter 10 of Gorenstein's book 'Finite
Groups'. He first proves the theorem that if f is any fixed-point-free
automorphism (i.e. f(x)=1 => x=1) of a finite group then, for any prime
p, f leaves a unique Sylow p-subgroup invariant Q, and any f-invariant
p-subgroup is contained in Q.
From this, in the case that f^3=Id, he proves that Q is in fact the
unique Sylow p-subgroup. The idea is that if x is a p-element not in Q,
then is f-invariant, contradicting the previous theorem.
In the following section, the more general result of Thompson (I believe
it was proved in Thompson's Doctoral thesis) that f fixed-point-free
of prime order implies G nilpotent is proved.
Derek Holt.