Some bismash products that are not group algebras. II. (Q372658)

Let \(S_n\) be the symmetric group in \(n\) symbols, \(n\geq 5\). Let also \(H_n=\mathbb C^C\#\mathbb CS\) be the (semisimple) Hopf algebra of dimension \(n!\), obtained as bismash product corresponding to the exact factorization of the symmetric group \(S_n=SC\), where \(S\) is the subgroup stabilizing \(n\) and \(C\) is the cyclic subgroup generated by the permutation \((1,2,\ldots,n)\).NEWLINENEWLINE The main result of the paper under review states that \(H_n\) is not isomorphic as an algebra to any group algebra. This result is related to Brauer's Problem 1 in finite group representation theory, that asks what \(\mathbb C\)-algebras are isomorphic to group algebras.NEWLINENEWLINE The main result extends the author's previous work in part I [J. Algebra 316, No. 1, 297-302 (2007; Zbl 1133.16027)]. The methods used in the paper under review, which are different from the ones employed in the mentioned previous work, involve, among others, the classification of finite simple groups, and make use of the unipotent characters of a simply connected group of Lie type.