A generalization of totally imprimitive permutation groups

Altınkaya, Rümeysa Sacide; Arıkan, Aynur

doi:10.22108/ijgt.2026.148234.2014

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	A generalization of totally imprimitive permutation groups
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 02 شهریور 1405 اصل مقاله (417.92 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2026.148234.2014
نویسندگان
Rümeysa Sacide Altınkaya^* ؛ Aynur Arıkan^*
Department of Mathematics, Faculty of Science, Gazi University, P.O.Box 06500, Ankara, T¨urkiye
چکیده
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.
کلیدواژه‌ها
Support-Block Property $(\mathcal{SBP})$؛ totally imprimitive group؛ permutation group؛ finitary permutation group

مراجع
[1] J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, New York, 1996. [2] P. Hall, Wreath powers and characteristically simple groups, Proc. Cambridge Philos. Soc., 58 (1962) 170–184. [3] B. Hartley and M. Kuzucuo˘glu, Non-simplicity of locally finite barely transitive groups, Proc. Edinburgh Math. Soc. (2), 40 no. 3 (1997) 483–490. [4] F. Leinen and O. Puglisi, Finitary representations and images of transitive finitary permutation groups, J. Algebra, 222 no. 2 (1999) 524–549. [5] D. McDougall, Soluble groups with the minimum condition for normal subgroups, Math. Z., 118 no. 3 (1970) 157–167. [6] H. Neumann, Varieties of Groups, Springer-Verlag, New York, 1967. [7] P. M. Neumann, On word subgroups of free groups, Arch. Math., 16 (1965) 6–21. [8] P. M. Neumann, The lawlessness of groups of finitary permutations, Arch. Math. (Basel), 26 no. 6 (1975) 561–566. [9] P. M. Neumann, The structure of finitary permutation groups, Arch. Math. (Basel), 27 no. 1 (1976) 3–17.
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A generalization of totally imprimitive permutation groups