Abstract:
All groups under consideration are finite. The paper studies some properties of the lattice of all τ -closed totally ω -saturated formations. We show that for any subgroup functor τ, the lattice of all τ -closed totally ω -saturated formations is modular and algebraic. We also prove that the lattice of all totally ω -saturated formations is G -separable. This strengthens a theorem of V.G. Safonov. Using embeddability the lattice of all τ -closed totally ω -saturated formations in the lattice of all totally ω -saturated formation, we establish that the lattice of all τ -closed totally ω -saturated formations is G−separable. In particular, we show that the lattice of all τ -closed totally ρ -saturated formations is modular, algebraic, and G−separable as well as the lattice of all τ -closed totally saturated formations.
Keywords:
formation of finite groups, totally ϖ-saturated formation, lattice of formations, τ-closed formation, modular lattice, algebraic lattice, separable lattice of formations.
Document Type:
Article
UDC:512.542
Language: Russian
Linking options:
https://www.mathnet.ru/eng/pmf44
This publication is cited in the following 6 articles:
Haiyan Li, A.-Ming Liu, Inna N. Safonova, Alexander N. Skiba, “Characterizations of some classes of finite
σ
-soluble
PσT
-groups”, Communications in Algebra, 52:1 (2024), 128
N. N. Vorob'ev, “On the modularity of the lattice of Baer-σ-local formations”, Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk, 59:1 (2023), 7
N. Yang, N. N. Vorob'ev, I. I. Staselka, “On the Modularity and Algebraicity of the Lattice of Multiply ω-Composition Fitting Classes”, Russ Math., 67:4 (2023), 66
N. N. Vorob'ev, “Modularity of the Lattice of Baer n-Multiply σ-Local Formations”, Algebra Logic, 62:4 (2023), 303
Aleksandr Tsarev, “Algebraic lattices of partially saturated formations of finite groups”, Afr. Mat., 31:3-4 (2020), 701
Aleksandr Tsarev, Andrei Kukharev, “Algebraic lattices of solvably saturated formations and their applications”, Bol. Soc. Mat. Mex., 26:3 (2020), 1003