FINITE GROUPS WITH WEAKLY S-SEMIPERMUTABLY EMBEDDED SUBGROUPS

Year 2012, Volume: 11 Issue: 11, 111 - 124, 01.06.2012

Zhencai Shen Jinshan Zhang Shulin Wu

Abstract

A subgroup H of G is said to be S-quasinormal in G if H permutes
with every Sylow subgroup of G. This concept was introduced by Kegel in
1962 and has been investigated by many authors. A subgroup H is called
S-semipermutable in G if H permutes with every Sylow p-subgroup of G for
which (p, |H|) = 1. A subgroup H of the group G is said to be c-normal in G
if there is a normal subgroup B of G such that HB = G and H ∩ B is normal
in G. Next, we unify and generalize the above concepts and give the following
concept: A subgroup H of the group G is said to be weakly S-semipermutably
embedded in G if there is a subnormal subgroup B of G such that HB = G
and H ∩ B is S-semipermutable or S-quasinormally embedded in G. Groups
with certain weakly S-semipermutably embedded subgroups of prime power
order are studied.

Keywords

weakly S-semipermutably embedded subgroup, p-nilpotent group, supersolvable group, formation

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• LMAM and School of Mathematical Sciences Peking University Beijing, 100871, P.R. China e-mail: [e-mail]
• Jinshan Zhang and Shulin Wu School of Science Sichuan University of Science and Engineering Zigong, 643000, P. R. China e-mails: [e-mail] (Jinshan Zhang) [e-mail] (Shulin Wu)