Vo
l.
32
( 2012 )
No.6
数学杂志
J.
of
Math.
(PRC)
A
CLASSIFICATION
OF
MINIMAL
NON-MSSP-GROUPS
GUO
Peng-fei
1
,2, AI
Yu-bo\
WANG
Jun-xin
3
(1.
Dept.
of
Mω
hematics
，
Lianyungang
Te
α
伽
r's
College,
Li
α
nyungang
222006,
Chin
α)
(2.
Shangh
ω
Institt
山
of
App
l.
Math. and
Mechani
叫
Shanghai
University,
Shangh
α
2000
切
，
Chin
α)
(3. Dept.
of
Mathemati
叫
Shanxi
Universit
ν
of
Finance and
E
ω
nomics
，
Taiyuan
030031
，
α
饥
α)
Abstract:
In this paper,
we
investigate the structure of a kind of finite minimal
non-~
groups. By analyzing the s-semipermutability
of
all minimal subgroups of the
Sylow
subgroups of
a group G
,
we
obtain a classification of a
new
kind of minimal
non-~-groups
，
which generalizes
some
achievements of minimal
non-~-groups.
Keywords:
MSSP-group; minimal non-MSSP-group; supersolvable group; power automor-
phism
2010
MR
Subject
Classification:
20D
lO;
20D20
Document
code: A
Article
ID:
0255-7797(2012)06-1005-06
1
Introduction
In
this
paper
,
only
且
nite
groups
are
considered
and
our
notation
is
standard.
Two
subgroups
H
and
K
of
a
group
G
are
said
to
permute
if H K = K
H.
A
subgroup
H
of a
group
G is called
qua
描蚓
s
剖
m
∞
10
町
r
口
rm
口
m
叫
(or
pe
曰
r
口
m
口
mutable
闷
e
时)
in
G if
it
pe
凹
r
口
rm
口
ml
丑lU川
te
臼
s
with
every
subgroup
of
G.
Kegel
[1]
called a
subgroup
H
of
a
group
G
s
叩
lasinormal
(or
s-permutable)
in G
if
it
permutes
with
every Sylow
subgroup
of
G. As a
gβneralization
of
s-permutability
,
Chen
[2]
introduced
the
following definition: a
subgroup
H
of
G is called
s
扑田
4
咽
s
盹
emin
∞
O
臼
rmτma
叫
1
in G
i
证
fH
pe
曰
r
口
rm
口
m
川
e
臼
s
with
every Sylow
p-
subgroup
of
G
with
(ω
p
，
IH
I)
= 1,
and
it
is
named
after
Zha口
I
归阴
3
叫]
s
萨
-s
回
emi
悟
pe
曰
r
口
rm
虹
nu
川
table
subgroup
of
G.
Studying
the
structure
of
groups
which
have
some
kind
of
property
plays
a key role
in
group
theory
and
many
吼叫
uable
results
about
this
topic
were
obtained.
For
example
,
Schmidt
[4]
determined
the
structure
of
minimal
non-nilpotent
groups
,
and
Doerk
[5]
deter-
mined
the
structure
of
minimal
non-supersolvable groups.
These
achievements
were
indeed
pushed
forward
the
developments
of
group
theoη.
The
further
results
can
consult
[6-10].
In
this
paper
, we call a
group
G
an
MSSP-group
if all
maximal
subgroups
of
the
Sylow
subgroups
of
G
are
s-semipermutable
in G, G a
minimal
non-MSSP-group
if
every
proper
•
Received
date:
2011-05-16
Accepted
date:
2012-04-11
Foundation
item:
Supported
by
National Natural
Science
Foundation
of
China(11071155); the
Overseas Study Project of Jiangsu Province
for
Outstanding
Young
and
Middl
e-
agedτ
国
chers
and Prin-
cipals
of
Colleges and
Unive
咽
ities([2011]34);
Qing Lan Project of Jiangsu Province
Biography:
Guo Pengfei(1972-), male, born
at
Wuxia
吨，
Shanxi, postdoctor, major
in
finite group
theory.
E-
mail:
guop
f9
99@163.com.
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