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Conjugacy Classes and Characters of Finite p-Groups

Let K be a conjugacy class of a finite p-group G where p is a prime, and let K −1 denote the conjugacy class of G consisting of the inverses of the elements in K. We observe that, in several cases, the number of elements in the product KK −1 is congruent to 1 modulo p − 1, and we pose the question in which generality this congruence is valid. We also consider related properties of the class multiplication constants of G. Furthermore, let χ be an irreducible character of G, and let denote the complex conjugate of χ. We show that, in several cases, the number of irreducible constituents of the product is congruent to 1 modulo p − 1, and we pose the question in which generality this congruence is valid.