Cyclic complementary extensions via automorphisms of generalized quaternion groups
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초록
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.
Abstract
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.
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