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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 KingdomP.J.Cameron@qmul.ac.uk
ALEXANDER W. DENT
Affiliation:
Information Security Group, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, United KingdomA.Dent@rhul.ac.uk
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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

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