2013/08/05 by E. V. Sokolov, Sokolov, E. V. · 1 citation
Mathematics · #20E06 #20F18 #FOS: Mathematics #Finite Group Theory Research #Functional Equations Stability Results #Group Theory (math.GR) #math.GR #msc:20E06 #msc:20E22 #msc:20E26 #msc:20F18 #primary 20E22 #secondary 20E26
paper · pdf · doi:10.48550/arxiv.1308.1039
This is an Author's Original Manuscript of an article submitted for consideration in the "Communications in Algebra" [copyright Taylor & Francis]; "Communications in Algebra" is available online at http://www.tandfonline.com
arxiv created 2013/08/05 · openalex publication_date 2013/08/05 · arxiv updated 2013/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A class of groups C is root in a sense of K. W. Gruenberg if it is closed under taking subgroups and satisfies the Gruenberg condition: for any group X and for any subnormal sequence Z \leqslant Y \leqslant X with factors in C, there exists a normal subgroup T of X such that T \leqslant Z and X/T ∈ C. We prove that a class of groups is root if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we prove also that, if C is a nontrivial root class of groups closed under taking quotient groups and G = <A*B; H=K, φ> is the generalized free product of two nilpotent C-groups A and B possessing φ-compartible central series, then G is residually a solvable C-group.