an:03987537
Zbl 0611.20003
Manz, Olaf; Staszewski, Reiner
On the number of generators and the modular group-ring of a finite \(p\)-group
EN
Proc. Am. Math. Soc. 98, 189-195 (1986).
00152086
1986
j
20C05 20D15 20C20 16S34 16Nxx 20F05
finite \(p\)-groups; Jacobson radical; group rings; descending central series; Loewy series; minimal number of generators
Let \(K\) be a field of characteristic \(p>0\) and let \(P\) be a finite \(p\)-group. Let \(JK(P)\) be the Jacobson radical of the group ring \(K(P)\). A particular descending central series, the Loewy series, is defined by letting \(\kappa_ 1(P)=P\), \(\kappa_ n(P)=[\kappa_{n-1}(P),P]\kappa_ m(P)^ p\) where m is the least integer for which \(pm\geq n\) (\(n=1,2,\dots,\ell+1)\) where \(\kappa_{\ell}(P)\neq 1\), \(\kappa_{\ell +1}(P)=1\). Let \(p^{d_ n}=| \kappa_ n(P):\kappa_{n+1}(P)|\) (\(n=1,2,\dots,\ell\)) and \(c_ i=\dim_ KJK(P)^ i/JK(P)^{i+1}\) \((i=0,1,...,s)\) where \(JK(P)^ s\neq 0\), \(JK(P)^{s+1}=0\). By the work of \textit{S. A. Jennings}
\[
\prod^{\ell}_{n=1}(1+t^ n+...+t^{n(p- 1)})^ n=\sum^{s}_{i=0}c_ i t^ i
\]
[Trans. Am. Math. Soc. 50, 175-185 (1941; Zbl 0025.24401)]. The Loewy series is said to be ``even monotonic'' if \(c_{i-1}\leq c_ i\) (1\(\leq i\leq s/2)\). It is shown, by example, that the Loewy series is not always even monotonic but that it is so if \(P\) is either Abelian, \(p\)-regular or extra-special. Noting that \(d=d_ 1=c_ 1\) is the minimal number of generators of \(P\) it is shown that \(c_ n\geq d\) (1\(\leq n\leq s-1)\), generalizing a result of \textit{B. K??lshammer} [J. Algebra 88, 190-195 (1984; Zbl 0567.20007)]. Other results of interest on the Loewy series are given.
D.A.R.Wallace
Zbl 0025.24401; Zbl 0567.20007