Finiteness conditions for the weak commutativity construction

2019/07/01 by Raimundo Bastos, Bastos, Raimundo, Bruno César Rodrigues Lima +3
Mathematics · #20E25 #20E34 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1907.00508

openalex publication_date 2019/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The operator, χ, of weak commutativity between isomorphic groups G and G^φ was introduced by Sidki as χ(G)=⟨ G ∪ G^φ| \lbrack g,g^φ]=1 ∀ g∈ G⟩ . It is known that the operator χ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove that if G is a locally finite group with exp(G)=n, then χ(G) is locally finite and has finite n-bounded exponent. Further, we examine some finiteness criteria for the subgroup D(G) = ⟨ [g₁,g₂^φ] | g_i ∈ G⟩ \leqslant χ(G) in terms of the set \[g₁,g₂^φ] | g_i ∈ G\.

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