On the subgroup separability of the free product of groups

2025/01/28 by E. V. Sokolov, Sokolov, E. V.
Mathematics · #20E06 #20E26 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2501.17342

openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that C is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), G is the free product of residually C-groups A_i (i ∈ I), and H is a subgroup of G satisfying a non-trivial identity. We prove a criterion for the C-separability of H in G. It follows from this criterion that, if \V_j | j ∈ J\ is a family of group varieties, each V_j (j ∈ J) is distinct from the variety of all groups, and V = \bigcup_{j ∈ J} V_j, then one can give a description of C-separable V-subgroups of G provided such a description is known for every group A_i (i ∈ I).

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