A remark on the Jacobson radical of a block ideal in a finite p-solvable group
Let F be a field of characteristic \(p>0\), G a finite p-solvable group, P a Sylow p-subgroup of G, \(p^ r\) (r\(\geq 2)\) the order of P, FG the group algebra of G over F and J(FG) the Jacobson radical of FG. Let t(G) be the index of nilpotency of J(FG). The author extends his own work for \(p\geq 5\) [J. Algebra 80, 134-144 (1983; Zbl 0507.16011)] and here proves that the following are equivalent: (1) \(t(G)=p^{r-1}+p-1\), (2) \(p^{r- 1}<t(G)<p^ r\), (3) P is noncyclic but has a cyclic subgroup of index p, (4) G has p-length 1 and P is isomorphic either to an Abelian group of type \((p^{r-1},p)\) or to a group of the form \(<a,b|\) \(a^ p=1\), \(b^{p^{r-1}}=1\), \(a^{-1}ba=b^ m>\). The proof depends on showing, by a complicated group-theoretic induction argument, that if \(t(G)>p^{r-1}\) then G has an element of order \(p^{r-1}\). It is observed that for \(p=2\) the main result does not hold.
scientific article; zbMATH DE number 3959687