Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T22:04:36.032Z Has data issue: false hasContentIssue false
  • English
  • Français

ORBIT-HOMOGENEITY IN PERMUTATION GROUPS

Published online by Cambridge University Press:  24 July 2006

PETER J. CAMERON  and
ALEXANDER W. DENT
PETER J. CAMERON
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United [email protected]
ALEXANDER W. DENT
Affiliation:
Information Security Group, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, United [email protected]
Get access

Abstract

This paper introduces the concept of orbit-homogeneity of permutation groups: a group $G$ is orbit-$t$-homogeneous if two sets of cardinality $t$ lie in the same orbit of $G$ whenever their intersections with each $G$-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of $t$-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan.

Further, this paper shows that any group generated by orbit-$t$-homogeneous subgroups is orbit-$t$-homogeneous, and that the condition becomes stronger as $t$ increases up to $\lfloor n/2\rfloor$, where $n$ is the degree. So any group $G$ has a unique maximal orbit-$t$-homogeneous subgroup $\Omega_t(G)$, and $\Omega_t(G)\le\Omega_{t-1}(G)$. Some structural results for orbit-$t$-homogeneous groups, and a number of examples, are also given.

Keywords

20B10
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 38 , Issue 4 , August 2006 , pp. 587 - 596
Copyright
The London Mathematical Society 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)