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For a finite group 
$G$, define 
$l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where 
$o(g)$ denotes the order of 
$g\in G$. We prove that if 
$l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or 
$l(G)>l(C_{2}\times C_{2})$, then 
$G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.
