On n-isoclinic embedding of groups

Abstract

By definition, the groups G and H are n-isoclinic ( n≥1) whenever G and H share the following properties: 1. (a) There is an isomorphism α from G ζ n(G) onto H ζ n(H) ; 2. (b) There is an isomorphism β from γ n+1 ( G) onto γ n+1 ( H); 3. (c) [ g 1,…, g n+1 ] α =[ h 1,…, h n+1 ] for any h i ∈( g i ζ n ( G)) α , g i ∈ G, i=1,…, n+1. It was known that 1-isoclinic groups G and H can be isomorphically embedded into a group X such that Gζ 1( X)= X= Hζ 1( X). In this paper it is investigated if such an embedding Gζ n ( X)= X= Hζ n ( X) is possible, if n≥2 and G and H are n-isoclinic. Some obstructions are encountered.