Maximal non-commuting sets in certain unipotent upper-triangular linear groups.

We find the exact size of a maximal non-commuting set in the unipotent upper triangular linear group $${UU_4(\mathbb {F}_q)}$$ in terms of a non-commuting geometric structure (refer to Definition 8), 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 strucuture. [ABSTRACT FROM AUTHOR]