On CLT and non-CLT groups (Q6624139)

A finite group \(G\) is called \(\mathsf{CLT}\)-group (converse of Lagrange's theorem) if it possesses subgroups of every possible order; otherwise it is non-\(\mathsf{CLT}\). It is well-known that \(\mathsf{CLT}\) groups are solvable and that supersolvable groups are \(\mathsf{CLT}\), however these two inclusions are proper.\N\NThe main result in the paper under review is Theorem 1.1: For every integer \(d \geq 2\) which is not a prime power, there exists a finite solvable group \(G\) such that \(d\) divides \(|G|\), \(\pi(G)=\pi(d)\) and \(G\) has no subgroup of order \(d\).\N\NLet \(D(G)\) denote the number of divisors \(d\) of \(|G|\) for which there exists a subgroup of \(G\) of order \(d\) and \(\tau(|G|)\) the number of all divisors of \(|G|\). The author defines the \(\mathsf{CLT}\)-degree of \(G\) as \(d_{\mathsf{CLT}}(G)=|D(G)|\cdot|\tau(G)|^{-1}\).\N\NLet \(\mathcal{G}\) be the class of all finite groups. The author proves that the set \(\{d_{\mathsf{CLT}}(G) \mid G \in \mathcal{G} \}\) is dense in \([0,1]\) and it is clear from the proof that the same conclusion holds for the class of all finite solvable groups.