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 $u \circ v \ne u*v$ 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 $G[\beta,\gamma,h]$).
@article{119414,
author = {Ale\v s Dr\'apal},
title = {Cyclic and dihedral constructions of even order},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {593-614},
zbl = {1101.20014},
mrnumber = {2062876},
language = {en},
url = {http://dml.mathdoc.fr/item/119414}
}
Drápal, Aleš. Cyclic and dihedral constructions of even order. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 593-614. http://gdmltest.u-ga.fr/item/119414/
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