Mersenne primes and solvable Sylow numbers (Q2816937)

Let \(G\) be a finite group and \(p\) a prime. The Sylow \(p\)-number \(n_p(G)\) of \(G\), i.e., the number of Sylow \(p\)-subgroups in \(G\), is called solvable if, for any prime \(l\), the \(l\)-part of \(|G|\) is congruent to 1 modulo \(p\). The Sylow theorem asserts that \(n_p(G)\equiv 1\pmod{p}\). \textit{P. Hall} [J. Lond. Math. Soc. 3, 98--105 (1928; JFM 54.0145.01)] showed that Sylow numbers of solvable groups are solvable and \textit{M. Hall jun.} [J. Algebra 7, 363--371 (1967; Zbl 0178.02102)] showed in the general case that \(n_p(G)\) is the product of two kinds of factors: of prime powers \(q^t\) with \(q^t\equiv 1\pmod{p}\) and of the Sylow \(p\)-number in certain finite simple groups (involved in \(G\)). These results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, the authors show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to \(L_2(q)\) for a Mersenne prime \(q\). This result solves a problem posed by \textit{J. Zhang} [J. Algebra 176, No. 1, 111--123 (1995; Zbl 0832.20042)].