Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T00:58:40.914Z Has data issue: false hasContentIssue false
  • English
  • Français

The group of almost automorphisms of the countable universal graph

Published online by Cambridge University Press:  24 October 2008

J. K. Truss
J. K. Truss
Affiliation:
Department of Pure Mathematics, University of Leeds
Get access Rights & Permissions [Opens in a new window]

Extract

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 223 - 236
Copyright
Copyright © Cambridge Philosophical Society 1989

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.)

References

REFERENCES

[1]Cameron, P. J.. Random structures, universal structures and permutation groups. (Handwritten notes.)Google Scholar
[2]Rubin, M.. On the reconstruction of ℵ0-categorical structures from their automorphism groups. (To appear.)Google Scholar
[3]Truss, J. K.. Embeddings of infinite permutation groups. In Proceedings of Groups – St Andreivs 1985, London Math. Soc. Lecture Note Series no. 121 (Cambridge University Press, 1986), pp. 335–351.Google Scholar
[4]Truss, J. K.. The group of the countable universal graph. Math. Proc. Cambridge Philos. Soc. 98 (1985), 213–245.CrossRefGoogle Scholar