Characterization of PSU (3,q) by its order and one special conjugacy class size

SupposethatGbeafinitegroup,andletN(G)be the set of conjugacy class sizes of G.  By Thompson’s conjecture,if H is a  finite  non  abelian  simple  group, G is  a  finite  group with  atrivialcenter,andN (G) = N (H),thenHandG   areisomorphic.Chen  et  al. contributed interestingly to Thompsons conjecture underaweakcondition.Inthisarticle,weinvestigatevalidityofThompsons conjecture under a weak condition for the projectivespecial unitary groups.This work implies that Thompsons conjectureholdsforthePS U(3,q),whereqisprimepower.2020MathematicsSubjectClassification:20D08,20D60