Pairwise‎ ‎non-commuting elements in finite metacyclic $2$-groups and some finite $p$-groups

Abstract

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ <br />‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ <br />‎we determine the maximum size of these subsets in any finite‎ <br />‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.