Lifting automorphisms of subgroups of direct products of cyclic $p$-groups

Dietz, Jill

doi:10.22108/ijgt.2024.142837.1924

تعداد نشریات	44
تعداد شماره‌ها	1,877
تعداد مقالات	15,278
تعداد مشاهده مقاله	43,730,738
تعداد دریافت فایل اصل مقاله	17,563,071

	Lifting automorphisms of subgroups of direct products of cyclic $p$-groups
International Journal of Group Theory
دوره 14، شماره 4، اسفند 2025، صفحه 235-252 اصل مقاله (481.8 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.142837.1924
نویسنده
Jill Dietz^*
Department of Mathematics, Statistics, and Computer Science, St. Olaf College Northfield, MN, USA
چکیده
Let $\Gamma$ be a finite group. A subgroup $H$ of $\Gamma$ is called ``fully liftable" in $\Gamma$ if every automorphism of $H$ is the restriction of an automorphism of $\Gamma$. Let $G=C_{p^{k_1}}\times C_{p^{k_2}}$, where $1\le k_1\le k_2$ and $p$ is prime. Using information about the subgroup structure of $G$ and knowledge of ${\rm Aut}(G)$, we characterize all fully liftable subgroups of $G$. It turns out that all cyclic subgroups of $G$ are fully liftable, and non-cyclic subgroups are fully liftable if and only if they are automorphic to certain subproducts of $G$, where two subgroups $H$ and $K$ are automorphic in $G$ if there exists $\alpha\in{\rm Aut}(G)$ such that $\alpha(H)=K$. Further, we compare the fully liftable subgroups of $G$ with the characteristic subgroups of $G$, which are similarly characterized by certain subproducts. Finally, we exhibit some interesting lattice features of both fully liftable subgroups of $G$ and characteristic subgroups of $G$.
کلیدواژه‌ها
Finite $p$-groups؛ automorphisms؛ characteristic subgroups

مراجع
[1] L. An, Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian p-group, Comm. Algebra, 50 no. 7 (2022) 2846–2853. [2] R. Baer, Der Kern, eine charakteristische Untergruppe. (German), Compositio Math., 1 no. 7 (1935) 254–283. [3] G. G. Bastos, E. Jespers, S. O. Juriaans and A. de A. e Silva, Extension of automorphisms of subgroups, Glasg. Math. J., 54 no. 2 (2012) 371–380. [4] Y. Chen, Y. Jiang and S. Jia, On the number of fuzzy subgroups of finite abelian p-groups, Int. J. Algebra, 6 no. 5-8 (2012) 233–238. [5] C. Y. Chew, A. Y. M. Chin and C. S. Lim, The number of subgroups of finite abelian p-groups of rank 4 and higher, Comm. Algebra, 48 no. 4 (2020) 1538–1547. [6] C. Conţiu, Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group, Carpathian J. Math., 27 no. 2 (2011) 193–199. [7] M. Golasiński and D. L. Gonçalves, On automorphisms of finite abelian p-groups, Math. Slovaca, 58 no. 4 (2008) 405–412. [8] M. Hampejs, N. Holighaus, L. Tóth and C. Wiesmeyr, Representing and Counting the Subgroups of the Group Zm × Zn , Journal of Numbers, 2014 no. 1 (2014). [9] C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly, 114 no. 10 (2007) 917–923. [10] A. Humam and P. Astuti, On the structure of characteristic subgroup lattices of finite abelian p-groups, Jordan J. Math. Stat., 15 no. 3A (2022) 435–444. [11] B. L. Kerby and E. Rode, Characteristic subgroups of finite abelian groups, Comm. Algebra, 39 no. 4 (2011) 1315–1343. [12] P. Kumar, Maximal cyclic subgroups of a finite abelian p-group of rank two, Quasigroups Related Systems, 28 no. 2 (2020) 237–242. [13] G. A. Miller, On the subgroups of an abelian group, Ann. of Math. (2), 6 no. 1 (1904) 1–6. [14] G. A. Miller, Determination of All the Characteristic Subgroups of any Abelian Group, Amer. J. Math., 27 no. 1 (1905) 15–24. [15] J. Oh, An explicit formula for the number of subgroups of a finite abelian p-group up to rank 3, Commun. Korean Math. Soc., 28 no. 4 (2013) 649–667. [16] A. N. Skiba, On finite groups for which the lattice of S-permutable subgroups is distributive, Arch. Math. (Basel), 109 no. 1 (2017) 9–17. [17] M. Tărnăuceanu, Contributions to the study of subgroup lattices, Matrix Rom, Bucharest, 2016. [18] M. Tărnăuceanu, Breaking points in centralizer lattices, C. R. Math. Acad. Sci. Paris, 356 no. 8 (2018) 843–845.
آمار تعداد مشاهده مقاله: 539 تعداد دریافت فایل اصل مقاله: 475

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

Lifting automorphisms of subgroups of direct products of cyclic $p$-groups