On finite soluble groups which behave like nilpotent groups with respect to the Frattini group
It is known that if \(G\) is a finite nilpotent group with subgroups \(H\) and \(K\), \(K\) normal in \(G\), then \(\Phi(H) \leq \Phi(G)\) and \(\Phi(G/K) = \Phi(G) \cdot K/K\). In this interesting note the author characterizes finite solvable groups which possess these \(\Phi\)-features. The nature of the results is indicated by Theorem \(1'\) (stated in the vernacular by the author). Let \(G\) be a finite solvable group. The following statements are equivalent: (a) If \(H\) and \(K\) are subgroups of \(G\) with \(H \leq K\) then \(\Phi(H) \leq \Phi(K)\). (b) \(G/\Phi(G)\) has elementary abelian Sylow subgroups. (c) If \(H/K\) is a \(p\)-chief factor of \(G\) then \(G/C_ G(H/K)\) is a \(p'\)-group with elementary abelian Sylow subgroups. (d) If \(H \leq G\) then \(\Phi(H) \leq \Phi(G)\). (e) If \(P\) is a Sylow subgroup of \(G\) then \(\Phi(P) \leq \Phi(G)\). In general the terminology and methods of the theory of classes of groups are used.
scientific article; zbMATH DE number 614222