Malaysian Journal of Mathematical Sciences, June 2023, Vol. 17, No. 2
The Exhaustion Numbers of the Generalized Quaternion Groups
Chen, H. V. and Sin, C. S.
Corresponding Email: chenhv@utar.edu.my
Received date: 13 June 2022
Accepted date: 6 February 2023
Abstract:
Let $G$ be a finite group and let $T$ be a non-empty subset of $G$. For any positive integer $k$, let $T^k=\{ t_1\dots t_k\mid t_1, \dots, t_k\in T\}$. The set $T$ is called exhaustive if $T^n=G$ for some positive integer $n$ where the smallest positive integer $n$, if it exists, such that $T^n=G$ is called the exhaustion number of $T$ and is denoted by $e(T)$. If $T^k\neq G$ for any positive integer $k$, then $T$ is a non-exhaustive subset and we write $e(T)=\infty$. In this paper, we investigate the exhaustion numbers of subsets of the generalized quaternion group $Q_{2^n}=\langle x$, $y\mid x^{2^{n-1}}=1$, $x^{2^{n-2}}=y^2$, $yx=x^{2^{n-1}-1}y\rangle$ where $n\ge 3$. We show that $Q_{2^n}$ has no exhaustive subsets of size $2$ and that the smallest positive integer $k$ such that any subset $T\subseteq Q_{2^n}$ of size greater than or equal to $k$ is exhaustive is $2^{n-1}+1$. We also show that for any integer $k\in\{3, \dots, 2^n\}$, there exists an exhaustive subset $T$ of $Q_{2^n}$ such that $\lvert T\rvert=k$.
Keywords: finite group; generalized quaternion group; exhaustion number
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