TITLE:
Note on Finite Groups Possessing a Normal Series with Factors of Sub-Class Size 1
AUTHORS:
Qi Liang, Yanheng Chen, Jiawei Li
KEYWORDS:
Finite Groups, Sub-Class Sizes, Factor Group, Solvable Group
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.11,
November
5,
2026
ABSTRACT: For a finite group
G
and any
x∈G
,
G
x
is defined to be the unique minimal subnormal subgroup of
G
that contains the element
x
. The conjugacy class of
x
in
G
x
is called the sub-class of
x
in
G
, and its size is called the sub-class size of
x
in
G
. This note mainly investigates finite groups that admit a normal series in which every factor group has sub-class size 1. It is proved that if each factor group of some normal series of a finite group
G
has sub-class size 1, then
G
is solvable. Conversely, if
G
is solvable, then there exists a normal series of
G
such that each of its factor groups has sub-class size 1. In particular,
G
is a solvable group if and only if each chief factor of
G
has sub-class size 1.