On the intersection of certain maximal subgroups of a finite group

✍ Scribed by Ballester-Bolinches, Adolfo; Beidleman, James C.; Heineken, Hermann; Ragland, Matthew F.; Schmidt, Jack

• Book ID: 125523848
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2014
• Tongue: English
• Weight: 200 KB
• Volume: 17
• Category: Article
• ISSN: 1433-5883
• DOI: 10.1515/jgt-2013-0052

✦ Synopsis

Abstract.

Let

Δ
(
G
)

${\Delta (G)}$

denote the intersection of all
non-normal maximal subgroups of a group G. We introduce
the class of
T
~2~-groups which are defined as the groups G for
which

G
/
Δ
(
G
)

${G/\Delta (G)}$

is a T-group, that is, a group in
which normality is a transitive relation. Several
results concerning the class T
~2~ are discussed. In
particular, if G is a
solvable group, then Sylow permutability is a transitive relation in G if and only if every subgroup H of G
is a T
~2~-group such that the nilpotent residual of
H is a Hall
subgroup of H.