Simple components and central units in group algebras.
Let \(G\) be a finite group and \(F\) be a field whose characteristic does not divide the order of \(G\). Denote by \(C_g\) the conjugacy class of \(g\in G\) and write \(\gamma_g=\sum_{h\in C_g}h\). Let \(e\) be the exponent of \(G\) and \(\theta\) a primitive \(e\)-th root of unity. For \(\sigma\in\text{Gal}(F(\theta),F)\) set \(g^\sigma=g^{i_\sigma}\), where \(i_\sigma\) is given by \(\sigma(\theta)=\theta^{i_\sigma}\). S. D. Berman and E. Witt independently determined the number of simple components of \(FG\) with \(\text{char\,}F=0\). Using a different approach, the author determines this number for arbitrary characteristic: it is shown that the number of simple components of \(FG\) is equal to the number of different orbits of the \(\gamma_g\)'s under the action of \(\text{Gal}(F(\theta),F)\). The author also computes the free rank of the group of the central units of \(\mathbb{Z} G\). This was independently done by \textit{J. Ritter} and \textit{S. K. Sehgal} by a different method [in Math. Proc. R. Ir. Acad. 105A, No. 1, 25-39 (2005)]. Furthermore, the author gives an alternative proof of a result by A. Giambruno and E. Jespers on the free rank of the group of the central units of \(\mathbb{Z} A_n\), where \(A_n\) is the alternating group on \(n\) letters. It is also determined for which \(n\) the central units of \(\mathbb{Z} A_n\) are trivial.
scientific article; zbMATH DE number 2117263
scientific article