Finite groups in which all maximal subgroups are $\pi$-closed. I
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 25-34
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Finite simple nonabelian groups $G$ that are not $\pi$-closed for some set of primes $\pi$ but have $\pi$-closed maximal subgroups (property $(*)$ for $(G,\pi)$) are studied. We give a list $\mathcal{L}$ of finite simple groups that contains any group $G$ with the above property (for some $\pi$). It is proved that $2\not\in\pi$ for any pair $(G,\pi)$ with property $(*)$ (Theorem 1). In addition, we specify for any sporadic simple group $G$ from $\mathcal{L}$ all sets of primes $\pi$ such that the pair $(G,\pi)$ has property $(*)$ (Theorem 2). The proof uses the author's results on the control of prime spectra of finite simple groups.
Keywords:
finite group; simple group; $\pi$-closed group; maximal subgroup; control of prime spectrum of a group.
@article{TIMM_2015_21_1_a2,
author = {V. A. Belonogov},
title = {Finite groups in which all maximal subgroups are $\pi$-closed. {I}},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {25--34},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {2015},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a2/}
}
V. A. Belonogov. Finite groups in which all maximal subgroups are $\pi$-closed. I. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 25-34. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a2/