On solvable groups with a minimal normal \(p\)-subgroup (Q1200935)

Let \(G\) be a finite solvable group, and \(p\) be a prime. Then the author defines \(G\) to be ``\(p\)-minimal'' if its Sylow \(p\)-subgroups are not normal and: (i) \(O_ p(O_{p'}(G))=O_ p(G)\times O_{p'}(G)\); and, (ii) \(| O_ p(G)|=p^{p-1}\) if \(p\) is a Fermat prime, and \(p^ p\) otherwise. The author proves a number of properties about finite solvable groups and about \(p\)-minimal groups in particular. Amongst these are the following: If \(G\) is \(p\)-minimal, then the Sylow \(p\)-subgroup of \(G/O_ p(G)\) has order \(p\); and \(O_ p(G)\) has a complement in \(G\). If \(p\) is an odd prime, then there are \(p\)-minimal groups of odd order exactly when \(p\) is not a Fermat prime. A Sylow \(p\)-subgroup of a \(p\)- minimal group has maximal class (class \(p-1\) if \(p\) is a Fermat prime and class \(p\) otherwise). If \(G\) is \(p\)-minimal with \(O_{p'}(G)=1\), then every subgroup of \(p\)-length 2 is also \(p\)-minimal.