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research
On
Ο
\sigma
Ο
-quasinormal subgroups of finite groups
Authors
Bin Hu
Jianhong Huang
Alexander N. Skiba
Publication date
28 January 2018
Publisher
View
on
arXiv
Abstract
Let
G
G
G
be a finite group and
Ο
=
{
Ο
i
β£
i
β
I
}
\sigma =\{\sigma_{i} | i\in I\}
Ο
=
{
Ο
i
β
β£
i
β
I
}
some partition of the set of all primes
P
\Bbb{P}
P
, that is,
Ο
=
{
Ο
i
β£
i
β
I
}
\sigma =\{\sigma_{i} | i\in I \}
Ο
=
{
Ο
i
β
β£
i
β
I
}
, where
P
=
β
i
β
I
Ο
i
\Bbb{P}=\bigcup_{i\in I} \sigma_{i}
P
=
β
i
β
I
β
Ο
i
β
and
Ο
i
β©
Ο
j
=
β
\sigma_{i}\cap \sigma_{j}= \emptyset
Ο
i
β
β©
Ο
j
β
=
β
for all
i
β
j
i\ne j
i
ξ
=
j
. We say that
G
G
G
is
Ο
\sigma
Ο
-primary if
G
G
G
is a
Ο
i
\sigma _{i}
Ο
i
β
-group for some
i
i
i
. A subgroup
A
A
A
of
G
G
G
is said to be:
Ο
{\sigma}
Ο
-subnormal in
G
G
G
if there is a subgroup chain
A
=
A
0
β€
A
1
β€
β―
β€
A
n
=
G
A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G
A
=
A
0
β
β€
A
1
β
β€
β―
β€
A
n
β
=
G
such that either
A
i
β
1
β΄
A
i
A_{i-1}\trianglelefteq A_{i}
A
i
β
1
β
β΄
A
i
β
or
A
i
/
(
A
i
β
1
)
A
i
A_{i}/(A_{i-1})_{A_{i}}
A
i
β
/
(
A
i
β
1
β
)
A
i
β
β
is
Ο
\sigma
Ο
-primary for all
i
=
1
,
β¦
,
n
i=1, \ldots, n
i
=
1
,
β¦
,
n
, modular in
G
G
G
if the following conditions hold: (i)
β¨
X
,
A
β©
Z
β©
=
β¨
X
,
A
β©
β©
Z
\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z
β¨
X
,
A
β©
Z
β©
=
β¨
X
,
A
β©
β©
Z
for all
X
β€
G
,
Z
β€
G
X \leq G, Z \leq G
X
β€
G
,
Z
β€
G
such that
X
β€
Z
X \leq Z
X
β€
Z
, and (ii)
β¨
A
,
Y
β©
Z
β©
=
β¨
A
,
Y
β©
β©
Z
\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z
β¨
A
,
Y
β©
Z
β©
=
β¨
A
,
Y
β©
β©
Z
for all
Y
β€
G
,
Z
β€
G
Y \leq G, Z \leq G
Y
β€
G
,
Z
β€
G
such that
A
β€
Z
A \leq Z
A
β€
Z
. In this paper, a subgroup
A
A
A
of
G
G
G
is called
Ο
\sigma
Ο
-quasinormal in
G
G
G
if
L
L
L
is modular and
Ο
{\sigma}
Ο
-subnormal in
G
G
G
. We study
Ο
\sigma
Ο
-quasinormal subgroups of
G
G
G
. In particular, we prove that if a subgroup
H
H
H
of
G
G
G
is
Ο
\sigma
Ο
-quasinormal in
G
G
G
, then for every chief factor
H
/
K
H/K
H
/
K
of
G
G
G
between
H
G
H^{G}
H
G
and
H
G
H_{G}
H
G
β
the semidirect product
(
H
/
K
)
β
(
G
/
C
G
(
H
/
K
)
)
(H/K)\rtimes (G/C_{G}(H/K))
(
H
/
K
)
β
(
G
/
C
G
β
(
H
/
K
))
is
Ο
\sigma
Ο
-primary.Comment: 9 page
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Last time updated on 17/10/2019