A condition for a finite group to be a Schmidt group

V. A. Belonogov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 81-86

Résumé

Let $G$ be a finite group $G$, and let $\pi$ be a set of primes such that $2\in \pi$. We prove that if all maximal subgroups of $G$ are $\pi$-closed and $G$ itself is not $\pi$-closed then $G$ is a Schmidt group. The proof employs the author's earlier results on the properties of pairs $(G,\pi)$ where $G$ is a simple minimal non-$\pi$-closed group and $\pi$ is arbitrary.

Détail

Keywords: finite group, Schmidt group, $\pi$-closed group, maximal subgroup.
Mots-clés : simple group

Bibliographie

[1] Miller G.A., Moreno H., “Nonabelian groups in which every subgroup is abelian”, Trans. Amer. Math. Soc., 1903, no. 4, 398–404 | DOI | MR | Zbl

[2] Shmidt O.Yu., “Gruppy, vse podgruppy kotorykh spetsialnye”, Mat. sb., 31:3–4 (1924), 366–372 | Zbl

[3] Shemetkov L.A., “O. Yu. Shmidt i konechnye gruppy”, Ukr. mat. zhurn., 23:5 (1971), 586–590 | Zbl

[4] Monakhov V.S., “O podgruppakh Shmidta konechnykh grupp”, Voprosy algebry, 13 (1998), 153–171

[5] Knyagina V.N., Monakhov V.S., “O konechnykh gruppakh s nekotorymi subnormalnymi podgruppami Shmidta”, Sib. mat. zhurn., 45:6 (2004), 1316–1322 | MR | Zbl

[6] Lutsenko Yu.V., Skiba A.N., “Stroenie grupp Shmidta i grupp Belonogova, v kotorykh lyubye dve 3-maksimalnye podgruppy yavlyayutsya $F(G)$-perestanovochnymi”, Vestn. Viteb. gos. un-ta. Matematika, 2009, no. 2 (52), 134–138 | MR | Zbl

[7] Tyutyanov V.N., “Konechnye gruppy s $\mathbb{P}$-subnormalnymi podgruppami Shmidta”, Problemy fiziki, matematiki i tekhniki, 22:1 (2015), 88–91 | Zbl

[8] Huppert B., Endliche gruppen, v. I, Springer, Berlin, 1967, 793 pp. | MR | Zbl

[9] Gorenstein D., Finite groups, Harper Row, N. Y., 1968, 527 pp. | MR | Zbl

[10] Chunikhina I.K., Chunikhin S.A., “O $p$-razlozhimykh gruppakh”, Mat. sb., 15:2 (1944), 325–342 | MR

[11] Arad Z., Chillag D., “A criterium for the existence of normal $\pi$-complements in finite groups”, J. Algebra, 87:2 (1984), 472–482 | DOI | MR | Zbl

[12] Belonogov V.A., “O kontrole prostogo spektra konechnoi prostoi gruppy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:3 (2013), 29–44 | MR

[13] Itô N., “Note on $(LM)$-groups of finite orders”, Kodai Math. Sem., 3:1–2 (1951), 1–6 | MR

[14] Belonogov V.A., “On finite minimal non-$\pi$-closed groups”, Abstracts of the Internat. Conf. and PhD-Master Summer School “Graphs and Groups, Spectra and Symmetries”, Sobolev Inst. Math., Novosibirsk, 2016, 47

[15] Belonogov V.A., “O konechnykh gruppakh, vse maksimalnye podgruppy kotorykh $\pi$-zamknuty”, Mezhdunar. shk.-konf. po teorii grupp, posvyasch. 70-letiyu V.V. Kabanova, sb. st., Kabard.-Balkar. gos. un-t., Nalchik, 2014, 6–9

[16] Belonogov V.A., “Some conditions for a finite group to be a Schmidt group”, XI shk.-konf. po teorii grupp, tez. dokl. Mezhdunar. konf., posvyasch. 70-letiyu A.Yu. Olshanskogo, Krasnoyarsk, 2016, 66–67

[17] Feit W., Thompson J.G., “Solvability of groups of odd order”, Pacific J. Math., 13 (1963), 755–1029 | MR