On the -elements of a finite group
Abstract: Let be a finite group, a prime, and a -element in . An element in is called a witness of if the subgroup generated by and is a -group. The set of all witnesses of in is denoted by . This paper shows that belongs to a given Sylow -subgroup of if one of the following holds: (1) is -solvable and ; (2) is -solvable, , and ; (3) and ; (4) normalizes a subgroup of with and ; (5) and .
Keywords: Largest solvable normal subgroup, center, nilpotent, nilpotent class, Fitting subgroup, Frattini subgroup, -solvable, simple group, socle, Sylow -subgroup, witness
Article copyright: © Copyright 1975 American Mathematical Society