A characterization of the symplectic and unitary 3-transposition groups

Abstract

Let D be a normal set of 3-transpositions in a group G. For each d ∈ D, let D_d be the set of elements ε in D such that the order of de equals 2. Then we can define an equivalence relation ⊺ on D by dre if and only if D_d ⋃ {d} = D_e ⋃ {ε}.

We characterize the symplectic and unitary groups over GF(2), respectively, GF(4) generated by their transvections as groups generated by a class D of 3-transposition with r trivial on D, but not when restricted to D_d for some d ∈ D.