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Generating the augmentation ideal. (English) Zbl 0782.20026
L. G. Kovács and Hyo-Seob Sim [Indag. Math., New Ser. 2, 229-232 (1991; Zbl 0751.20028)] have shown that if a finite group $$G$$ can be generated by subgroups $$H_ 1,H_ 2,\dots,H_ s$$, each of which can be generated by $$r$$ elements and if $$| H_ i|$$ are pairwise coprime, then $$G$$ is solvable and can be generated by $$r+s-1$$ elements and also a similar result if $$[G:H_ i]$$ are pairwise coprime. In this paper, the authors prove if $$G$$ is generated by subgroups $$H_ 1,H_ 2,\dots,H_ s$$, $$| H_ i|$$ are pairwise coprime and if the augmentation ideal of $$\mathbb{Z} H_ i$$ can be generated as $$H_ i$$ modules by $$r$$ elements; then the augmentation ideal of $$\mathbb{Z} G$$ can be generated by $$r+s-1$$ element. And a similar result if $$[G:H_ i]$$ are pairwise coprime. As a corollary follow the results of Kovács and Hyo- Seob Sim.

MSC:
 20F05 Generators, relations, and presentations of groups 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
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References:
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