On Π-property of subgroups of a finite group

Abstract

Let [Formula: see text] be a finite group, [Formula: see text] a prime, [Formula: see text] a Sylow [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] a power of [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the unique smallest normal subgroup of [Formula: see text] for which the corresponding factor group is abelian of exponent dividing [Formula: see text]. Let [Formula: see text], [Formula: see text], [Formula: see text] be classes of all [Formula: see text]-groups, [Formula: see text]-nilpotent groups and [Formula: see text]-supersolvable groups, respectively, [Formula: see text] be the [Formula: see text]-residual of [Formula: see text]. Let [Formula: see text]. A subgroup [Formula: see text] of a finite group [Formula: see text] is said to have [Formula: see text]-property in [Formula: see text], if for any [Formula: see text]-chief factor [Formula: see text], [Formula: see text] is a [Formula: see text]-number. A normal subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-hypercyclically embedded in [Formula: see text] if every [Formula: see text]-[Formula: see text]-chief factor of [Formula: see text] is cyclic, where [Formula: see text] is a fixed prime. In this paper, we prove that [Formula: see text] is [Formula: see text]-hypercyclically embedded in [Formula: see text] if and only if for some [Formula: see text]-subgroups [Formula: see text] of [Formula: see text], [Formula: see text] have [Formula: see text]-property in [Formula: see text].