# Article

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Keywords:
cyclic construction; dihedral construction; quarter distance
Summary:
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 $G[\beta,\gamma,h]$).
References:
[1] Bálek M., Drápal A., Zhukavets N.: The neighbourhood of dihedral $2$-groups. submitted.
[2] Donovan D., Oates-Williams S., Praeger C.E.: On the distance of distinct Latin squares. J. Combin. Des. 5 (1997), 235-248. MR 1451283
[3] Drápal A.: Non-isomorphic $2$-groups coincide at most in three quarters of their multiplication tables. European J. Combin. 21 (2000), 301-321. MR 1750166
[4] Drápal A.: On groups that differ in one of four squares. European J. Combin. 23 (2002), 899-918. MR 1938347 | Zbl 1044.20009
[5] Drápal A.: On distances of $2$-groups and $3$-groups. Proceedings of Groups St. Andrews 2001 in Oxford, to appear. MR 2051524
[6] Drápal A., Zhukavets N.: On multiplication tables of groups that agree on half of columns and half of rows. Glasgow Math. J. 45 (2003), 293-308. MR 1997707
[7] Zhukavets N.: On small distances of small $2$-groups. Comment. Math. Univ. Carolinae 42 (2001), 247-257. MR 1832144 | Zbl 1057.20018

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