p-Hypercyclically embedding and Π-property of subgroups of finite groups

ABSTRACT

Let G be a finite group, E a normal subgroup of G and p a fixed prime. We say that E is p-hypercyclically embedded in G if every p-G-chief factor of E is cyclic. A subgroup H of G is said to satisfy Π-property in G if |G∕K:N_G∕K((H∩L)K∕K)| is a π((H∩L)K∕K)-number for any chief factor L∕K in G; we say that H has Π*-property in G if H∩O^π(H)(G) has Π-property in G. In this paper, we prove that E is p-hypercyclically embedded in G if and only if some classes of p-subgroups of E have Π*-property in G. Some recent results are extended.

KEYWORDS:

• p-Hypercyclically embedded subgroup
• Π-normality
• Π-property
• Π*-property

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D15

Acknowledgment

The authors wish to thank the help of Professor A. Ballester–Bolinches.