Algebraic lattices of solvably saturated formations and their applications

Abstract

In each group G, we select a system of subgroups \(\tau (G)\) and say that \(\tau\) is a subgroup functor if \(G \in \tau (G)\) for every group G, and for every epimorphism \(\varphi : A \rightarrow B\) and any \(H \in \tau (A)\) and \(T \in \tau (B)\), we have \(H^\varphi \in \tau (B)\) and \(T^{\varphi ^{-1}} \in \tau (A)\). We consider only subgroup functors \(\tau\) such that for any group G all subgroups of \(\tau (G)\) are subnormal in G. For any set of groups \({\mathfrak {X}}\), the symbol \({s}_\tau ({\mathfrak {X}})\) denotes the set of groups H such that \(H \in \tau (G)\) for some group \(G \in {\mathfrak {X}}\). A formation \({\mathfrak {F}}\) is \(\tau\)-closed if \({s}_\tau ({\mathfrak {F}}) = {\mathfrak {F}}\). The Frattini subgroup \({\varPhi }(G)\) of a group G is the intersection of all maximal subgroups of G. A formation \({\mathfrak {F}}\) is said to be solvably saturated if it contains each group G with \(G/{\varPhi }(N) \in {\mathfrak {F}}\) for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all \(\tau\)-closed totally composition formations is algebraic.

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Acknowledgements

We thank the anonymous referee for carefully reading the manuscript.

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Correspondence to Aleksandr Tsarev.

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This work has been supported by the Russian Science Foundation under Grant 18-71-10007.

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Tsarev, A., Kukharev, A. Algebraic lattices of solvably saturated formations and their applications. Bol. Soc. Mat. Mex. 26, 1003–1014 (2020). https://doi.org/10.1007/s40590-020-00290-3

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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