| Title:
|
Cyclic and dihedral constructions of even order (English) |
| Author:
|
Drápal, Aleš |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
593-614 |
| . |
| Category:
|
math |
| . |
| 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 $G[\beta,\gamma,h]$). (English) |
| Keyword:
|
cyclic construction |
| Keyword:
|
dihedral construction |
| Keyword:
|
quarter distance |
| MSC:
|
05B15 |
| MSC:
|
20D15 |
| MSC:
|
20D60 |
| idZBL:
|
Zbl 1101.20014 |
| idMR:
|
MR2062876 |
| . |
| Date available:
|
2009-01-08T19:31:29Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119414 |
| . |
| Reference:
|
[1] Bálek M., Drápal A., Zhukavets N.: The neighbourhood of dihedral $2$-groups.submitted. |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[4] Drápal A.: On groups that differ in one of four squares.European J. Combin. 23 (2002), 899-918. Zbl 1044.20009, MR 1938347 |
| Reference:
|
[5] Drápal A.: On distances of $2$-groups and $3$-groups.Proceedings of Groups St. Andrews 2001 in Oxford, to appear. MR 2051524 |
| Reference:
|
[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 |
| Reference:
|
[7] Zhukavets N.: On small distances of small $2$-groups.Comment. Math. Univ. Carolinae 42 (2001), 247-257. Zbl 1057.20018, MR 1832144 |
| . |
