On the p-Elements of a Finite Group
Copyright policy )On the p-Elements of a Finite Group
Let G be a finite group, p a prime, and x a p-element in G. An element g in G is called a witness of G if the subgroup generated by x and g is a p-group. The set of all witnesses of x in G is denoted by W(x). This paper shows that x belongs to a given Sylow p-subgroup P of G if one of the following holds: (1) G is p-solvable and W(x) Pfl n Ixgg E GI; (2) G is p-solvable, P = (P\Z(P)), and W(x)vP\Z(P); (3) cl(P)<2 and W(x) DP; (4) x normalizes a subgroup P1 of P with |P: P11 c p2 and W(x)DP; (5) IPI = P4 and W(x) D P.
Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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