A remark on the Loewy-layers of the modular group algebra of a finite p- group
Let G be a group of order \(p^ n\) and class n-1, and let J be the augmentation ideal of KG, where K is a field of characteristic p. For \(r\geq 0\), let \(c_ r\) be the dimension of \(J^ r/J^{r+1}\) over K. The dimension of \(\kappa_ nG/\kappa_{n+1}G\) is easily found from the structure of G, and a formula of Jennings then enables the \(c_ r\) to be calculated. This is used to show by hand that if \(p^ n=3^ 5\), then \(c_{10}\geq c_{11}\leq c_{12}\geq c_{13}\), and by computer that if \(p^ n=5^ 7\), \(c_{50}\geq c_{51}\leq c_{52}\). Using a theorem of Quillen, it is shown that the Lie algebras of these groups have the corresponding property.
scientific article; zbMATH DE number 3981412
scientific article