On the Frattini subgroup of a finite group (Q335604)

A group \(G\) is said to be \textit{\(\Phi\)-free} if its Frattini subgroup \(\Phi(G)\) is trivial. The main result of the paper under review shows that if \(G\) is a finite \(\Phi\)-free group such that \(G/N\) is \(\Phi\)-free for each normal subgroup \(N\) containing the generalized Fitting subgroup \(F^*(G)\), then all homomorphic images of \(G\) are \(\Phi\)-free. As a consequence, the authors obtain a generalization of a result by \textit{K. Doerk} [J. Algebra 167, No. 2, 533--537 (1994; Zbl 0806.20021)] on the structure of a finite soluble group \(G\) such that \(\Phi(G/N)=\Phi(G)N/N\) for every normal subgroup \(N\) of \(G\). Moreover, the authors prove that if \(G\) is a finite \(nC\)-group (i.e. a group in which every normal subgroup has a complement), then all subnormal subgroups of \(G\) are \(nC\)-groups. This is a positive answer to a long-standing question by \textit{C. Christensen} [Math. Z. 84, 52--69 (1964; Zbl 0134.02703)].