On Π-permutable subgroups in finite groups - Hu, B.; Huang, J.; Adarchenko, N. M. (Ukrains’kyi Matematychnyi Zhurnal, 2026)
UDC 512.542 Let σ ={σ_i | iI} be some partition of the set of all primes P and let Π be a nonempty subset of the set σ. A set H of subgroups of a finite group G is said to be a complete Hall Π-set of G if every member of H is a Hall σ _i-subgroup of G for some σ _iΠ and H contains exactly one Hall σ_i-subgroup of G for every σ _iΠ such that σ_iπ(G) eq. A subgroup A of G is called (i) H^G-permutable if AH^x=H^xA for HH and xG; (ii) Π-permutable in G if A is H^G-permutable for some complete Hall Π-set H of G. We study the influence of Π-permutable subgroups on the structure of G. In particular, we prove that if π= olimits_σ_iΠ σ_i and G =AB, where A and B are H^G-permutable π-separable (respectively, π-closed) subgroups of G, then G is also π-separable (respectively, π-closed). Some known results are generalized.