On generating a finite group by nilpotent subgroups (Q1346188)

The paper was motivated by a question that \textit{L. Babai} [Can. Math. Bull. 17, 467-470 (1974; Zbl 0311.05120)] had asked. It is shown that, for every positive integer \(n\), there exists a finite soluble group \(G_ n\) which can be generated by \(n\) but not fewer than \(n\) of its nilpotent subgroups. Groups satisfying this condition are constructed inductively. Given their important result, the authors then raise the problem to determine, for each positive integer \(N\), the smallest positive integer \(\nu(N)\) such that all groups of order \(N\) can be generated by at most \(\nu (N)\) of their nilpotent subgroups. The construction of the paper also provides the result that \(\nu (N)\) is less than or equal to the number of distinct prime divisors of \(N\).