If all the Sylow subgroups of a finite group \(G\) can be generated by \(d\) elements, then \(G\) itself can be generated by \(d+1\) elements.
This result has been proved independently by R. M. Guralnick in [Arch. Math. 53, No. 6, 521–523 (1989; Zbl 0675.20026)] and by the author in [Arch. Math. 53, No. 4, 313–317 (1989; Zbl 0679.20028)].
Given a sequence \(x={(x_n)}_{n \in \mathbb{N}}\) of independent uniformly distributed \(G\)-valued random variables, \(\tau_G=\min\{n \geq 1 \;| \;G= \langle x_1, x_2, \ldots, x_n \rangle\}\) denotes the minimum number of elements for the random generation of \(G\) and \(e(G)\) denotes the expectation of the random variable \(\tau_G\). A probabilistic version of the above result can be formulated in terms of \(e(G)\) (see Theorem 1):
If all the Sylow \(p\)-subgroups (\(p\) prime) of (a finite group) \(G\) can be generated by \(d\) elements, then \(e(G) \leq d + \eta\), where \(\eta = \frac{5}{2} + \sum_{p \geq 3} \frac{1}{{(p-1)}^2}<3\).
An accurate estimation of the term \(\sum_{p \geq 3} \frac{1}{{(p-1)}^2}\) is possible, thanks to a recent result of H. Cohen [“High precision computation of Hardy-Littlewood constants”, Preprint]; in fact \(\eta \simeq 2.875065\dots\).
As corollary (see Corollary 2), the author obtains that if \(G\) is a permutation group of degree \(n\), then \(e(G) \leq \lfloor n/2 \rfloor + \eta\).
Finally, the presence of the Haar measure allows an important extension of the original result to the infinite case, considering profinite groups, namely if all the Sylow subgroups of a profinite group \(G\) are topologically \(d\)-generated, then \(G\) is topologically \((d+1)\)-generated and \(e(G) \leq d + \eta\). A weaker formulation of this result is presented by the final Theorem 3.
The proof of the main result (Theorem 1) is divided into three lemmas, which use a series of important concepts of number theory, combinatorics, probabilistic group theory, and co-homology.