A note on locally graded minimal non-metahamiltonian groups


Abstract: We prove that a nonperfect locally gradedminimal non-metahamiltonian group $G$ is a soluble group withderived length of at most 4. On the other hand, if $G$ is perfect,then $G/\Phi (G)$ is isomorphic to $A_{5}$, where $\Phi (G)$ isthe Frattini subgroup of $G$ and $A_{5}$ 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.

Keywords: Locally graded minimal non-metahamiltonian, soluble, metabelian, $d$-maximal,

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