A characterization of the groups L3(2n)

This note is concerned with finite groups in which a Sylow two-subgroup S has an elementary Abelian subgroup E of order 2²ⁿ, n≥2, such that E=A × z(S), ¦A¦=2ⁿ, and C_S(a)=E for any involutiona ∈ A. It is proved that a simple group satisfying this condition is isomorphic to L₃,(2ⁿ).