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An investigation of countable B-groups

Published online by Cambridge University Press:  24 October 2008

P. Cameron  and
K. W. Johnson
P. Cameron
Affiliation:
Merlon College, Oxford
K. W. Johnson
Affiliation:
Mathematics Department, Pennsylvania State University, Ogontz Campus, Abington, PA 19001, U.S.A.
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Extract

A group G is defined to be a B-group if any primitive permutation group which contains G as a regular subgroup is doubly transitive. In the case where G is finite the existence of families of B-groups has been established by Burnside, Schur, Wielandt and others and led to the investigation of S-rings. A survey of this work is given in [3], sections 13·7–13·12. In this paper the possibility of the existence of countable B-groups is discussed. Three distinct methods are given to embed a countable group as a regular subgroup of a simply primitive permutation group, and in each case a condition on the square root sets of elements of the group is necessary for the embedding to be carried out. It is easy to demonstrate that this condition is not sufficient, and the general question remains open.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 223 - 231
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Erdös, P. and Spencer, J.. Probabilistic Methods in Combinatorics (Academic Press, 1984).Google Scholar
[2]Truss, J. K.. The group of the countable universal graph. Math. Proc. Cambridge Philos. Soc. 98 (1985), 213–245.CrossRefGoogle Scholar
[3]Scott, W. R.. Group Theory (Prentice-Hall, 1964).Google Scholar