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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

##### MSC:
 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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