Finite groups are pervasive in mathematics.In studies into the structure of a finite group, the action of a finite group A on a finite group G such that the orders \((|A|\, {\rm and}\, |G|)\) of A and G are coprime is a basic topic. Here we present a Theorem in this topic which extends basic results “from a single prime to a finite set of primes” and implies that for any finite group G, if A acts trivially on \(F^*(G)\) (the generalized Fitting subgroup of G), then A acts trivially on G. Also our Theorem implies and extends a useful result of Z. Arad and G. Glauberman and implies and extends Thompson’s celebrated \(P \times Q\) Lemma and an important Lemma of D. M. Goldschmidt.
