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

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Statistics, and Computer Science, St. Olaf College Northfield, MN, USA

Abstract

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 $G$.

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References

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International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 04 October 2024

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History

  • Receive Date: 17 September 2024
  • Revise Date: 30 September 2024
  • Accept Date: 04 October 2024
  • Published Online: 04 October 2024

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