Abstract:Let $G(\circ )$ and $G(*)$ be two groups of finite order $n$, and suppose that they share a normal subgroup $S$ such that $u\circ v = u *v$ if $u \in S$ or $v \in S$. Cases when $G/S$ is cyclic or dihedral and when $u \circ v \not =u*v$ for exactly $n^2/4$ pairs $(u,v) \in G\times G$ have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible $G(*)$ from a given $G = G(\circ )$. The constructions, denoted by $G[\alpha ,h]$ and $G[\beta ,\gamma ,h]$, respectively, depend on a coset $\alpha $ (or two cosets $\beta $ and $\gamma $) modulo $S$, and on an element $h \in S$ (certain additional properties must be satisfied as well). The purpose of the paper is to expose various aspects of these constructions, with a stress on conditions that allow to establish an isomorphism between $G$ and $G[\alpha ,h]$ (or $G[\beta ,\gamma ,h]$).
Keywords: cyclic construction, dihedral construction, quarter distance
AMS Subject Classification: Primary 20D60; Secondary 05B15