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Bull. Korean Math. Soc. 2024; 61(5): 1269-1287

Online first article September 25, 2024      Printed September 30, 2024

https://doi.org/10.4134/BKMS.b230537

Copyright © The Korean Mathematical Society.

Cyclic complementary extensions via automorphisms of generalized quaternion groups

Kan Hu , Manyu Zhang

Muzejski trg 2; Zhejiang Ocean University

Abstract

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A cyclic complementary extension of a finite group $A$ is a finite group $G$ which
contains $A$ as a subgroup and contains a cyclic subgroup $C$ such that $A\cap C=\{1_G\}$ and
$G=AC$.
A skew morphism of a finite group $A$ is a permutation $\varphi$ on $A$ such
that  $\varphi(1_A)=1_A$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$,
where $\pi:A\to\mathbb{Z}_{|\varphi|}$
is a power function associated with $\varphi$. For  a positive multiple $n$ of the order $|\varphi|$ of $\varphi$,
if there exists an extended power function $\Pi:A\to\mathbb{Z}_n$ of $\varphi$, then $\varphi$ and $\Pi$ can be
used to construct a cyclic complementary extension of $A$. In this paper, we use this approach to
construct and classify the cyclic complementary extensions of the generalized quaternion groups
corresponding to automorphisms.

Keywords: Skew morphism, group factorization, solvable group

MSC numbers: Primary 20B25, 05E18, 57M15

Supported by: This work was financially supported by the Slovenian Research Agency (ARRS) and the research project N1-0208.

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