Semisimple metacyclic group algebras. (Q1930297)

There has been intensive research recently on central idempotents of group algebras of finite groups, such as by \textit{G. K. Bakshi} and \textit{I. B. S. Passi}, [in Commun. Algebra 40, No. 4, 1413-1426 (2012; Zbl 1254.20003)]. Let \(G\) be a group of order \(p_1p_2\) with \(p_1\) and \(p_2\) distinct primes, \(F\) a Galois field of order \(q\) such that the group algebra \(FG\) is semisimple. The authors determine a complete system of primitive central idempotents and the Wedderburn decompoisiton of the algebra \(FG\). As a consequence, the automorphism group of the group algebra is described. Moreover, as examples, the general results are applied for some particular groups \(G\).