Generating the augmentation ideal (Q2366589)

\textit{L. G. Kovács} and \textit{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.