Cyclic and dihedral constructions of even order

Let G(◦) and G(∗) be two groups of finite order n, and suppose that they share a normal subgroup S such that u ◦ v = u ∗ v if u ∈ S or v ∈ S. Cases when G/S is cyclic or dihedral and when u ◦ v = u ∗ v for exactly n2/4 pairs (u, v) ∈ G× 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(◦). The constructions, denoted by G[α, h] and G[β, γ, h], respectively, depend on a coset α (or two cosets β and γ) modulo S, and on an element h ∈ 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[α, h] (or G[β, γ, h]).