Finite groups and their coprime automorphisms

Abstract

Let p p be a prime and A A a finite group of exponent p p acting by automorphisms on a finite p ′ p’ -group G G . Assume that A A has order at least p 3 p^3 and C G ( a ) C_G(a) is nilpotent of class at most c c for any a ∈ A # a\in A^{\#} . It is shown that G G is nilpotent with class bounded solely in terms of c c and p p .

Fitting Subgroup and Nilpotent Residual of Fixed Points

“…Subsequently the above result was extended to the case where A is not necessarily abelian. Namely, it was shown in [2] that if A is a finite group of prime exponent q and order at least q 3 acting on a finite q ′ -group G in such a manner that C G (a) is nilpotent of class at most c for any a ∈ A # , then G is nilpotent with class bounded solely in terms of c and q. Many other results illustrating the influence of centralizers of automorphisms on the structure of G can be found in [7].…”

On groups with automorphisms whose fixed points are Engel

“…It was recently shown in [13] that the assumption that A is abelian is actually superfluous in the above result. That is, the result holds for any group A of prime exponent.…”

Nilpotent residual and fitting subgroup of fixed points in finite groups

“…only". In the recent article [3] the above result was extended to the case where A is not necessarily abelian. Namely, it was shown that if A is a finite group of prime exponent q and order at least q 3 acting on a finite q ′ -group G in such a manner that C G (a) is nilpotent of class at most c for any a ∈ A # , then G is nilpotent with class bounded solely in terms of c and q.…”

“…# . An alternative way of expressing this is to say that C G (B) is generated by the subgroups C G (A i ) (see for example [3,Lemma 2.3]) . Now, for any subgroup…”