Cyclic and dihedral constructions of even order

Aleš Drápal

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 593-614
  • ISSN: 0010-2628

Abstract

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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 n 2 / 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 ] ).

How to cite

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Drápal, Aleš. "Cyclic and dihedral constructions of even order." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 593-614. <http://eudml.org/doc/249199>.

@article{Drápal2003,
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 $u \circ v \ne u*v$ 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 $G[\beta ,\gamma ,h]$).},
author = {Drápal, Aleš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cyclic construction; dihedral construction; quarter distance; finite 2-groups; Hamming distance; group multiplication tables; Cayley tables},
language = {eng},
number = {4},
pages = {593-614},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cyclic and dihedral constructions of even order},
url = {http://eudml.org/doc/249199},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Drápal, Aleš
TI - Cyclic and dihedral constructions of even order
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 593
EP - 614
AB - 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 $G[\beta ,\gamma ,h]$).
LA - eng
KW - cyclic construction; dihedral construction; quarter distance; finite 2-groups; Hamming distance; group multiplication tables; Cayley tables
UR - http://eudml.org/doc/249199
ER -

References

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  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. (1997) MR1451283
  3. Drápal A., Non-isomorphic 2 -groups coincide at most in three quarters of their multiplication tables, European J. Combin. 21 (2000), 301-321. (2000) MR1750166
  4. Drápal A., On groups that differ in one of four squares, European J. Combin. 23 (2002), 899-918. (2002) Zbl1044.20009MR1938347
  5. Drápal A., On distances of 2 -groups and 3 -groups, Proceedings of Groups St. Andrews 2001 in Oxford, to appear. MR2051524
  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. (2003) MR1997707
  7. Zhukavets N., On small distances of small 2 -groups, Comment. Math. Univ. Carolinae 42 (2001), 247-257. (2001) Zbl1057.20018MR1832144

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