Self dual groups of order \(p^5\) (\(p\) an odd prime)
Let \(G\) be a finite group, let \(\text{Irr}(G)=\{\chi_1,\dots,\chi_k\}\) be the set of irreducible complex characters and let \(\text{Cl}(G)=\{C_1,\dots,C_k\}\) be the set of conjugacy classes of \(G\). For each \(i\) fix \(x_i\in C_i\). A group \(G\) is called self dual, if \(| C_j|\chi_i(x_j)/\chi_i(1)=\chi_j(1)\chi_j(x_j)\) for all \(i,j\). In particular \(| C_j|=\chi_j(1)^2\). The interest in such groups comes from a connection between \(Z(\mathbb{C} G)\), the center of \(\mathbb{C} G\), and fusion algebras in mathematical physics. This concept has been introduced by \textit{E. Bannai} [in J. Algebr. Comb. 2, No. 4, 327-344 (1993; Zbl 0790.05098)]. Unfortunately self dual groups seem to be rare. By a result of T. Okuyama (still unpublished) self dual groups are nilpotent with self dual Sylow subgroups. Thus a classification of self dual groups reduces to the classification of self dual \(p\)-groups. Non-abelian \(p\)-groups of order \(\leq p^4\) are not self dual. In the note under review, Hanaki shows that for \(p\) odd self dual groups of order \(p^5\) exist. Actually, he proves that all groups in the isoclinism class \(\Phi_6\) in the classification table of groups of order \(p^5\) [\textit{R. James}, Math. Comput. 34, 613-637 (1988; Zbl 0428.20013)]\ are self dual.
scientific article; zbMATH DE number 1080372