A note on locally graded minimal non-metahamiltonian groups

Abstract

We prove that a nonperfect locally graded minimal non-metahamiltonian group G is a soluble group with derived length of at most 4. On the other hand, if G is perfect, then G/Φ(G) is isomorphic to A5 , where Φ(G) is the Frattini subgroup of G and A5 is the alternating group. Moreover, we show that under some conditions, if G is a p -group, then G is metabelian, where p is a prime integer.