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On nearly \( \mathcal{M} \)-supplemented primary subgroups of finite groups. (English) Zbl 1417.20004

Summary: A subgroup \( H \) is called to be nearly \( \mathcal{M} \)-supplemented in \( G \) if \( G \) has a normal subgroup \( K \) such that \( HK\unlhd G \) and \( H_iK<HK \) for every maximal subgroup \( H_i \) of \( H \). The main goal of this paper is to investigate the structure of chief factors of finite groups by using nearly \( \mathcal{M} \)-supplemented primary subgroups and obtain some new characterization about chief factors of finite groups. The main result is the following: Let \( G\in E_{p'} \) and \( P \) be a Sylow \( p \)-subgroup of \( G \), where \( p \) is an odd prime. If every maximal subgroup of \( P \) is nearly \( \mathcal{M} \)-supplemented in \( G \), then every non-abelian \( pd \)-\( G\)-chief factor \( A/B \) satisfies one of the following conditions:
(1)
\( A/B \cong \mathrm{PSL} (2,7) \) and \( p=7; A/B \cong\mathrm{PSL} (2,11) \) and \( p=11 \);
(2)
\( A/B \cong \mathrm{PSL} (2,2^t) \) and \( p=2 ^t+1>3 \) is a Fermat prime;
(3)
\( A/B \cong \mathrm{PSL} (n,q) \), \( n\geq 3 \) is a prime, \( (n,q-1)=1 \) and \( p=q^n-1/q-1\);
(4)
\( A/B \cong M_{11} \) and \( p=11; A/B\cong M_{23}\) and \(p=23\);
(5)
\( A/B \cong A_p\) and \(p\geq 5\).

MSC:

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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