A generalization of totally imprimitive permutation groups

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Gazi University, P.O.Box 06500, Ankara, T¨urkiye

Abstract

In the present paper, we investigate a general form of totally imprimitive finitary permutation groups. Totally imprimitive groups were originally defined by P. M. Neumann for subgroups of the finitary symmetric group acting on an infinite set. We aim to extend this definition to arbitrary subgroups of symmetric groups, which are not necessarily finitary. To this end, we introduce a property that we shall call the support-block property for subgroups of symmetric groups. This property allows us to generalize several results previously obtained for totally imprimitive finitary permutation groups to a broader class of permutation groups acting on infinite sets. We recover and extend various structural results obtained in earlier studies, with particular emphasis on transitive permutation groups. Moreover, we reconstruct an example arising in previous studies on finitary permutation groups and show that it satisfies the support-block property, illustrating the applicability of our approach. Several consequences and applications of this framework are also discussed.

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References

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International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 24 August 2026

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History

  • Receive Date: 02 February 2026
  • Revise Date: 25 July 2026
  • Accept Date: 29 July 2026
  • Published Online: 24 August 2026

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