The Chern classes modulo \(p\) of a regular representation (Q1291220)

Let \(r_G\) be a complex regular representation of a finite group \(G\) and \(c_i(r_G)\) the mod \(p\) Chern class [\textit{M. F. Atiyah}, Publ. Math., Inst. Hautes Etud. Sci. 9, 247-288 (1961; Zbl 0107.02303)] of the representation \(r_G\). \textit{B. B. Venkov} [Dokl. Akad. Nauk SSSR 137, 1274-1277 (1961; Zbl 0116.25903)] announced that \(c_i(r_G)=0\) for \(i<p^n-p^{n-1}\), where \(p^n\) is the highest power of a prime \(p\) dividing the order \(|G|\). The author makes use of his methods presented earlier [J. Algebra 144, No. 1, 214-247 (1991; Zbl 0777.20019)] to show that this assertion is valid for any finite group \(G\).