Abstract
Let n be a positive integer, and let \(\sigma = \{\sigma _i \mid i \in I\}\) be a partition of the set of all primes. It is shown that every law of the lattice of all formations is fulfilled in the lattice of all n-multiply \(\sigma \)-local formations of finite groups. This result implies that the lattice of all n-multiply \(\sigma \)-local formations of finite groups is modular but not distributive for any nonnegative integer n. Let \({\mathfrak {F}}\) and \({\mathfrak {H}}\) be n-multiply \(\sigma \)-local formations of finite groups such that \({\mathfrak {H}} \subseteq {\mathfrak {F}}\). Denote by \({\mathfrak {F}} /_n^\sigma {\mathfrak {H}}\) the lattice of all n-multiply \(\sigma \)-local formations \({\mathfrak {M}}\) such that \({\mathfrak {H}} \subseteq {\mathfrak {M}} \subseteq {\mathfrak {F}}\). If \({\mathfrak {M}} \subset {\mathfrak {F}}\) and the lattice \({\mathfrak {F}} /_n^\sigma {\mathfrak {M}}\) consists of only two elements then \({\mathfrak {M}}\) is called a maximal n-multiply \(\sigma \)-local subformation of \({\mathfrak {F}}\). The properties of the intersection of the maximal n-multiply \(\sigma \)-local subformations of \(\mathfrak F\) are described.
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This work has been partially supported by Basic Science Research Program to RIBS of Jeju National University through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2019R1A6A1A10072987) and by the grant F19RM-071 from the Belarusian Republican Foundation for Fundamental Research.
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Tsarev, A. Laws of the Lattices of \(\sigma \)-Local Formations of Finite Groups. Mediterr. J. Math. 17, 75 (2020). https://doi.org/10.1007/s00009-020-01510-w
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DOI: https://doi.org/10.1007/s00009-020-01510-w
Keywords
- Finite group
- formation of groups
- \(\sigma \)-function
- n-multiply \(\sigma \)-local formation
- lattice of formations
- modular lattice
- law of the lattice
- maximal subformation
- Frattini subformation
- regular language
- formation of languages
- monoids in computer programming