Algebraic lattices of solvably saturated formations and their applications
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ORIGINAL ARTICLE
Algebraic lattices of solvably saturated formations and their applications Aleksandr Tsarev1,2 • Andrei Kukharev2,3 Received: 1 December 2019 / Accepted: 28 March 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In each group G, we select a system of subgroups sðGÞ and say that s is a subgroup functor if G 2 sðGÞ for every group G, and for every epimorphism u : A ! B and 1 any H 2 sðAÞ and T 2 sðBÞ, we have H u 2 sðBÞ and T u 2 sðAÞ. We consider only subgroup functors s such that for any group G all subgroups of sðGÞ are subnormal in G. For any set of groups X, the symbol ss ðXÞ denotes the set of groups H such that H 2 sðGÞ for some group G 2 X. A formation F is s-closed if ss ðFÞ ¼ F. The Frattini subgroup UðGÞ of a group G is the intersection of all maximal subgroups of G. A formation F is said to be solvably saturated if it contains each group G with G=UðNÞ 2 F for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all s-closed totally composition formations is algebraic. Keywords Finite group Subgroup functor Formation of groups Satellite of formation Totally composition formation Algebraic lattice of formations Formal language Hypergroup
Mathematics Subject Classification Primary 20F17 Secondary 20D10 43A62 20M35
This work has been supported by the Russian Science Foundation under Grant 18-71-10007. & Aleksandr Tsarev [email protected] Andrei Kukharev [email protected] 1
Department of Mathematics, Jeju National University, Jeju 690-756, South Korea
2
Department of Mathematics and IT, P.M. Masherov Vitebsk State University, 33 Moscow Avenue, 210038 Vitebsk, Belarus
3
Siberian Federal University, 79 Svobodny Avenue, 660041 Krasnoyarsk, Russia
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A. Tsarev, A. Kukharev
1 Introduction, definitions and the main result A variety of groups may be defined as a nonempty class of groups closed under taking homomorphic images and subcartesian products [22], formations extend this notion. Definition 1 [11] A formation is a class of finite groups F satisfying the following two conditions: (1) (2)
if G 2 F, then G=N 2 F, and if G=N1 , G=N2 2 F, then G=N1 \ N2 2 F,
for any normal subgroups N, N1 and N2 of G. The theory of saturated formations introduced by Gaschu¨tz [13] in 1962 became a fundamental part of group theory by now. Further research showed that formations are of general algebraic nature and can be applied to the study of infinite groups, Lie algebras, universal algebras and even of general algebraic systems, see, e.g., the books [3, 11, 12, 15, 29, 42]. Recall that the Frattini subgroup UðGÞ of a group G is the intersection of all maximal subgroups of G. Definition 2 [11] A formation F is said to be saturated if G=UðGÞ 2 F implies G 2 F. A well-known result states that any formation is saturated iff it is local [11, Gaschu¨tz–Lubeseder–Schmid]. This nice circumstance makes saturated formations one of the most suitable classes for a better understanding of group structure
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