Let G a finite group. For conjugacy classes A and B of G, the subset AB of G is a union of conjugacy classes of G, and for irreducible complex characters χ and φ{symbol} of G, the product χφ{symbol} is a linear combination of irreducible characters of G with positive integer coefficients. In this article we give the structure of finite groups G whenever C ~2 contains a central conjugacy class or does not contain a central conjugacy class of G, for all conjugacy classes C of G. Also in the case of (C ~(-1))~m C ~n ? Z(G) the structure of G is given. Similar results when C is replaced by an irreducible character χ of G are discussed.
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