THE NUMBER OF PAIRWISE NONCOMMUTING SETS IN A FINITE GROUP

We say that two nonempty subsets A and B with cardinality r of a group G are noncommuting subsets if $xy\neq yx$ for every $x\in A$ and $y\in B$ . We say a nonempty set $\mathcal {X}$ of subsets with cardinality r of G is an r-noncommuting set if every two elements of $\mathcal {X}$ are noncommuting subsets. If $|\mathcal {X}| \geq |\mathcal {Y}|$ for any other r-noncommuting set $\mathcal {Y}$ of G, then the cardinality of $\mathcal {X}$ (if it exists) is denoted by $w_G(r)$ and is called the r-clique number of G. In this paper, we try to find the influence of the function $w_G: \mathbb {N} \longrightarrow \mathbb {N}$ on the structure of groups.