Automorphisms and Fitting factors of finite groups

AbstractIt is proved that for every prime p, there exists a function fp such that, if G is a finite solvable group with an automorphism α of order p, then G contains a nilpotent normal subgroup of index at most fp(¦CG(α)¦). The question is first reduced to a representation theoretic problem which is then analyzed along the lines of the well-known Theorem B of Hall and Higman. Using an elementary argument of Brauer and Fowler, it is shown further that in the case p = 2, the assumption of solvability may be omitted. Thus, there is a function f such that for any group G of even order and any involution x in G, the Fitting factor group GF(G) has order at most f(¦CG(x)¦)