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Lifting automorphisms of subgroups of direct products of cyclic $p$-groups | ||
International Journal of Group Theory | ||
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 13 مهر 1403 اصل مقاله (481.72 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. [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). | ||
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