On recognizability of PSU_3(q) by the orders of maximal abelian subgroups

Abstract: In [Li and Chen, A new characterization of the simple group A_1(p^n), Sib. Math. J., 2012], it is proved that the simple group A_1(p^n) is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as A_2(q), Monatsh. Math., 2013], the authors proved that if L=A_2(q), where q is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as L, is isomorphic to L or an extension of L by a subgroup of the outer automorphism group of L. In this paper, we prove that if L=PSU_3(q), where q is not a Fermat prime, then every finite group with the same orders of maximal abelian subgroups as L, is isomorphic to L or an extension of L by a field automorphism of L.