On Non-Commuting Sets in a Finite $p$-Group with Derived Subgroup of Prime Order

Abstract

Let $G$ be a finite group. A nonempty subset $X$ of $G$ is said to be non-commuting if $xy≠yx$ for any $x, y ∈ X$ with $x≠y$. If $|X| ≥ |Y|$ for any other non-commuting set $Y$ in $G$, then $X$ is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite $p$-group with derived subgroup of prime order.