Poset-blowdowns of generalized quaternion groups

Sasaki, Kenjiro; Takamura, Shigeru; Hirakawa, Ryota

doi:10.22108/ijgt.2023.137628.1841

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	Poset-blowdowns of generalized quaternion groups
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 20 تیر 1402 اصل مقاله (10.68 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.137628.1841
نویسندگان
Kenjiro Sasaki^* ¹؛ Shigeru Takamura¹؛ Ryota Hirakawa²
¹Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, JAPAN
²Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, JAPAN
چکیده
Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: \textit{tame, wild}, and \textit{hybrid} depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups $Q_{2^n}$ $(n \geq 3)$. They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of $Q_{2^n}$.
کلیدواژه‌ها
Subgroup poset؛ Hasse diagram؛ poset map؛ Frattini subgroup؛ chief series

مراجع
[1] Y. Chen and G. Chen, A note on a characterization of generalized quaternion 2-groups, C. R. Math. Acad. Sci. Paris, 352 no. 6 (2014) 459–461. [2] K. Conrad, Generalized quaternions, (2010), https://kconrad.math.uconn.edu/blurbs/grouptheory/genquat.pdf [3] R. Schmidt, Subgroup lattices of groups, de Gruyter Expositions in Mathematics, 14, Walter de Gruyter GmbH & Co. KG, Berlin, 1994. [4] S. Takamura, Blowdown Maps between Subgroup Posets, Tokyo J. Math., 45 no. 2 (2022) 467–499. [5] M. Tarnauceanu, A characterization of generalized quaternion 2-groups, C. R. Math. Acad. Sci. Paris, 348 no. 13–14 (2010) 731–733. [6], Breaking points in the poset of conjugacy classes of subgroups of a finite group, Czechoslovak Math. J., 69 no. 4 (2019) 1081–1087.
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Poset-blowdowns of generalized quaternion groups