The Structure of Groups with Quasi-Primitive Characters Behnam Razzaghmaneshi 1 Abstract A finite group G is a QP-group if every irreducible character is quasi- primitive.In this paper we study the structure of QP-groups and the end show that A group G is a non-abelian QP-groups if and only if there ex- ist quasisimple subgroups P 1 , P 2 , ···, P n of G such that (i) G =(P 1 · ... · P n )Z(G), (ii) each P i is a QP-group, (iii) [P i , P j ]= 1, and P i ⊆ Z(P j ) whenever 1 ≤ i < j ≤ n, (iv) G is a covering group of G/Z(G), (v) [P 1 , ..., P n ]= 1, (vi) P 1 · ... · P n ⊆ Z n 2 − (n−1)(n−2) 2 (P n ), (vii) G =(P 1 · ... · P n )Z(G) ⊆ Z n 2 − (n−1)(n−2) 2 (P n ), (viii) [P i , P i ]= 1, for i = 1, ..., n, (ix) G ∼ P 1 · ... · P n , (x) If G is a QP-group and N  G, then N and G/N is also QP-group, (xi) Let G = 1 be a group. Then G is a QP-group with Z(G)= 1 if and only if G is a direct product of non-abelian simple QP-groups, (xii) If G is quasisimple, then the covering group G is Qp-groups. Key words and phrases: quasi-primitive group, quasisimple, Irreducible complex character. MSC(2010): 20F45; 20F05, 11Y35. 1 Introduction One of the most interesting problem in the study complex character theory of a fi- nite group G determined and classification primitive or quasi-primitive subgroups 1 Behnam Razzaghmaneshi, Department of Mathematics, Talessh Branch, Islamic Azad university, Talesh, Iran b razzagh@yahoo.com 1