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.