On Finite Groups with an Abelian Sylow Group

Abstract

We shall consider finite groups of order of g which satisfy the following condition: (*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group of then the centralizer (P) of P in coincides with the centralizer ( ) of in . This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions. Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).