When the Wedderburn decomposition of the semisimple group algebra FqG implies that of Fq(G × C2)?

Abstract

In this short note we give a condition under which the Wedderburn decomposition (WD) of the semisimple group algebra \(F_q(G\times C_2)\) can be directly deduced from the WD of the semisimple group algebra \(F_qG\), where \(F_q\) is a finite field with char\((F_q)>2\), \(G\) is an arbitrary finite group and \(C_2\) is a group of order 2. To complement the abstract theory with an example, we determine the WD of the semisimple group algebra \(F_q(A_5\times C_2)\), where \(A_5\) is the alternating group from that of \(F_qA_5\).