On the number of subgroups of a finite group (Q6601227)

For a finite group \(G,\) let \(s(G)\) denote the number of subgroups of \(G\) and \(d(G)\) denote the number of divisors of the order of \(G\). In 1984 \textit{I. M. Richards} [Am. Math. Mon. 91, No. 9, 571--572 (1984; \url{doi:10.1080/00029890.1984.11971498})] proved that \(s(G)\geq d(G)\) for any finite group \(G\), with equality if and only if \(G\) is cyclic. The authors refine this result, proving that if \(G\) is noncyclic, then \(s(G)\geq d(G)+p,\) where \(p\) is the smallest prime divisor of \(|G|\). Moreover they determine the structure of finite groups \(G\) such that \(s(G) = d(G) + p.\)