On groups whose small-order elements generate a small subgroup. (Q1929766)

The following theorem is proved. Theorem. Let \(G\) be a finite group, \(\pi(G)=\{p_1,\dots,p_k\}\) and \(b_1,\dots,b_k\in\mathbb N\cup\{0\}\). Then there exists a finite group \(K\) and \(N\triangleleft K\) such that (a) \(N'=\{1\}\) and \(K/N\cong G\). (b) \(\pi(K)=\pi(G)\). (c) Any element of \(K\) of order \(p_i^c\), where \(1\leq i\leq k\) and \(0\leq c\leq b_i\), belongs to \(N\). It follows from this that there exists a nonsolvable finite group \(G\) such that all elements of \(G\) of prime orders and of order \(4\) belong to \(\Phi(G)\).