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On the number of generators and the modular group-ring of a finite \(p\)-group. (English) Zbl 0611.20003
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 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 B. Külshammer [J. Algebra 88, 190-195 (1984; Zbl 0567.20007)]. Other results of interest on the Loewy series are given.
Reviewer: D.A.R.Wallace

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D15 Finite nilpotent groups, \(p\)-groups
20C20 Modular representations and characters
16S34 Group rings
16Nxx Radicals and radical properties of associative rings
20F05 Generators, relations, and presentations of groups
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