Weak classification of finite groups

Abstract

The purpose of this paper is to study groups (G, ⋅) with prescribed properties on the associated right loops (G, ○n) where ○n is the binary operation on G given by a ○n b = b-nabn+1, n being a fixed integer. Let us call a group G1 to be weakly n-isomorphic to a group G2 if there is a bijective map ϕ : G1 → G2 such that ϕ(y-nxyn+1) = ϕ(y)-nϕ(x)ϕ(y)n+1, ∀ x, y ∈ G1. Indeed, we initiate a weak classification program of finite groups and obtain some basic motivating results in this direction.