Basic 3-Transpositions of Unitary Group Un (2 )

We aim to study maximal pairwise commuting sets of 3-transpositions (transvections) of the simple unitary group [Formula: see text] over [Formula: see text], and to construct designs from these sets. Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions. Let [Formula: see text]. It is well known that [Formula: see text] is a 3-transposition group with the set [Formula: see text], the conjugacy class consisting of its transvections, as the set of 3-transpositions. Let [Formula: see text] be a set of basic transpositions in [Formula: see text]. We give general descriptions of [Formula: see text] and [Formula: see text]- [Formula: see text] designs [Formula: see text], with [Formula: see text] and [Formula: see text]. The parameters [Formula: see text], [Formula: see text] and further properties of [Formula: see text] are determined. We also, as examples, apply the method to the unitary simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].