Irreducible subrepresentations of the conjugation representation of finitep-groups

For finitep-groupsG we study the conjugation representation γG which is defined by lettingG act on itself by conjugation. Roth conjectured that every irreducible representation ofG which is trivial on the center ofG, occurs in γG. However, this is not true in general. We construct minimal counterexamples and verify Roth's conjecture for various classes of finitep-groups, for instance those of maximal class. Dedicated to Professor George Maltese on the occasion of his 60th birthday