Irreducible subrepresentations of the conjugation representation of finite \(p\)-groups
A finite group \(G\) acting on its group algebra \(\mathbb{C} G\) by conjugation gives rise to the conjugation representation \(\gamma_ G\) of \(G\). It was conjectured by \textit{R. L. Roth} [Pac. J. Math. 36, 515-521 (1971; Zbl 0192.357)] that \(\gamma_ G\) contains every irreducible representation of the central quotient group \(G/Z(G)\). However, using the existence of groups with outer automorphisms that preserve the conjugacy classes, \textit{E. Formanek} [Proc. Am. Math. Soc. 30, No. 3, 73-74 (1971; Zbl 0206.314)] constructed counterexamples. Let \(p\) be an arbitrary prime number. The authors construct counterexamples of order \(p^ 6\) and class 3 and of order \(p^ 7\) and class 2. They also show that these are minimal in the sense that a group of order \(p^ n\) and class \(c\) satisfies Roth's conjecture when \(n<6\) or when \(n=6\) and \(c\leq 2\).
scientific article; zbMATH DE number 60079