2022/02/03 by E. V. Sokolov, Sokolov, E. V. · 1 citation
Mathematics · #20E08 (Secondary) #20E26 #20F18 (Primary) 20E06 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2202.01378
openalex publication_date 2022/02/03 · openalex created_date 2022/04/03 · openalex updated_date 2026/08/04
Suppose that C is a class of groups consisting only of periodic groups and \mathfrakP(C)′ is the set of prime numbers each of which does not divide the order of any element of a C-group. A subgroup Y of a group X is called a) C-separable in this group if, for each x ∈ X ∖ Y, there exists a homomorphism σ of X onto a group from C such that xσ∉ Yσ; b) \mathfrakP(C)′-isolated in X if, for any x ∈ X, q ∈ \mathfrakP(C)′, the inclusion xq ∈ Y implies that x ∈ Y. It is easy to see that if Y is C-separable in X, then it is \mathfrakP(C)′-isolated in this group. Let us say that X has the property C-\mathfrakSep if all its \mathfrakP(C)′-isolated subgroups are C-separable. We find a condition that is sufficient for a nilpotent group N to have the property C-\mathfrakSep provided C is a root class (i.e., it contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian products of the form ∏v ∈ VUv, where U, V ∈ C and Uv is an isomorphic copy of U for each v ∈ V). We also prove that if N is torsion-free, then the indicated condition is necessary for this group to have C-\mathfrakSep.