The orders of subgroup products and coset products

Takamura, Shigeru

doi:10.22108/toc.2024.141562.2176

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	The orders of subgroup products and coset products
Transactions on Combinatorics
مقاله 3، دوره 14، شماره 4، اسفند 2025، صفحه 223-250 اصل مقاله (2.18 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2024.141562.2176
نویسنده
Shigeru Takamura^*
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, JAPAN
چکیده
A sect is a subset of a group given by the product of a finite number of subgroups. It is generally not a direct product nor even a subgroup of the group. For finite groups, the orders of sects are their basic invariants. In this paper we describe properties of the orders of sects, such as divisibility and inequalities, which give constraints on the possible values of the orders of sects. We further consider clans, which are subsets of groups given by products of finite numbers of cosets. We also describe properties of their orders.
کلیدواژه‌ها
Subgroup product؛ Coset product؛ Group factorization؛ Order inequality؛ Order divisibility

مراجع
[1] B. Amberg, S. Franciosi and F. Giovanni, Products of groups, Products of groups, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. [2] B. Amberg and L. Kazarin, Factorizations of groups and related topics, Sci. China Ser. A, 52 no. 2 (2009) 217–230. [3] M. Asaad, A. Ballester-Bolinches and R. Esteban-Romero, Products of finite groups, De Gruyter Expositions in Mathematics, 53, Walter de Gruyter GmbH & Co. KG, Berlin, 2010. [4] R. Hirakawa, K. Sasaki and S. Takamura, Poset-blowdowns of generalized quaternion groups, Int. J. Group Theory, 13 no. 2 (2024) 133–160. [5] S. Takamura, Classification of finite groups with birdcage-shaped Hasse diagrams, Osaka J. Math., 58 no. 4 (2021) 885–897. [6] S. Takamura, Blowdown maps between subgroup posets, Tokyo J. of Math., 45 no. 2 (2022) 467–499. [7] S. Takamura, Prime factorizations of finite groups, I, II, and III, to appear in Advances in Group Theory and Applications, (2023).
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The orders of subgroup products and coset products