Maximal Non-commuting Sets in Certain Unipotent Upper-triangular Linear Groups

We find the exact size of a maximal non-commuting set in unipotent uppertriangular linear group $U{U}_4({\mathbb{F}}_q)$ in terms of a non-commuting geometric structure (Refer Definition [10]), where ${\mathbb{F}}_q$ is the finite field with $q$ elements. Then we get bounds on the size of such a set by explicitly finding certain non-commuting sets in the non-commuting structure.