On weak commutativity in 𝑝-groups

Abstract

Abstract The weak commutativity group χ ⁢ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We present new bounds for the exponent of χ ⁢ ( G ) \chi(G) and its sections, when 𝐺 is a finite 𝑝-group.