CLTσ-group with σ-subnormal or Self-normalizing Subgroups
Let G be a finite group and σ = { σ i ∣ i ⋃ I } be a partition of the set of all primes ℙ, that is, ℙ = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j . A set ℋ of subgroups of G is said to be a complete Hall σ -set of G if every non-identity member of ℋ is a Hall σ i -subgroup of G for some i and ℋ co...
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Veröffentlicht in: | Frontiers of Mathematics 2023-07, Vol.18 (4), p.961-975 |
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Sprache: | eng |
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Zusammenfassung: | Let
G
be a finite group and
σ
= {
σ
i
∣
i
⋃
I
} be a partition of the set of all primes ℙ, that is, ℙ = ∪
i
∈
I
σ
i
and
σ
i
∩
σ
j
= ∅ for all
i
≠
j
. A set
ℋ
of subgroups of
G
is said to be a complete Hall
σ
-set of
G
if every non-identity member of
ℋ
is a Hall
σ
i
-subgroup of
G
for some
i
and
ℋ
contains exactly one Hall
σ
i
-subgroup of
G
for every
σ
i
∈
σ
(
G
). A group
G
is said to be a CLT
σ
-group if
G
has a complete Hall
σ
-set {
H
1
, …,
H
t
} such that for all subgroups
A
i
≤
H
i
,
G
has a subgroup of order ∣
A
1
∣ … ∣
A
t
∣. A group
G
is a generalized CLT
σ
-group, if
G
has a complete Hall
σ
-set {
H
1
, …,
H
t
} such that for all subgroups
A
i
≤
H
i
,
G
has a subgroup
A
of order ∣
A
1
∣ … ∣
A
t
∣ such that
A
satisfies some
σ
-normality condition. In this paper, we study the generalized CLT
σ
-groups which are characterized completely under some conditions. |
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ISSN: | 2731-8648 1673-3452 2731-8656 1673-3576 |
DOI: | 10.1007/s11464-020-0126-8 |