The ‘permutizer’ of a subgroup \(H\) of a group \(G\) is the subgroup generated by all cyclic subgroups of \(G\) which permute with \(H\). A group \(G\) is said to satisfy the ‘maximal permutizer condition’ when the permutizer of every maximal subgroup coincides with the whole group.
The main result proved in the paper is that a finite soluble group with \(\Phi(G)=1\) satisfies the maximal permutizer condition if and only if every minimal normal subgroup \(N\) of \(G\) either has prime order or has order \(4\) and \(G/C_G(N)\cong S_3\) (Theorem 1.3). As a corollary (Corollary 1.4), it is obtained that a finite group \(G\) with the maximal permutizer condition is supersoluble or else \(G/F(G)\) is supersoluble and \(F(G)/\Phi(G)\) is a direct product of \(G\)-chief factors of prime orders or order \(4\); moreover, if \(T\) is a \(G\)-chief factor between \(\Phi(G)\) and \(F(G)\) of order \(4\), then \(G/C_G(T)\cong S_3\).
Finally, in Theorem 2.4 it is proved that under certain hypotheses the maximal permutizer condition is preserved by taking some subdirect products. More precisely, if \(G\) is a finite group with \(\Phi(G)=1\), \(M\) and \(N\) are two distinct minimal normal subgroups of \(G\) such that \(G/M\) and \(G/N\) both satisfy the maximal permutizer condition, then \(G\) satisfies the maximal permutizer condition.