# Restriction of characters and products of characters - Mathematics > Group Theory

Abstract: Let G be a finite p-group, for some prime p, and $\psi, \theta \in \IrrG$be irreducible complex characters of G. It has been proved that if, inaddition, $\psi,\theta$ are faithful characters, then the product $\psi\theta$is a multiple of an irreducible or it is the nontrivial linear combination ofat least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that ifwe do not require the characters to be faithful, then given any integer k>0, wecan always find a p-group G and irreducible characters $\Psi$ and $\Theta$ suchthat $\Psi\Theta$ is the nontrivial combination of exactly k distinctirreducible characters. We do this by translating examples of decompositions ofrestrictions of characters into decompositions of products of characters.