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A new criterion for solvability of a finite group by the sum of orders of non-normal subgroups
Let $G$ be a finite group and $\nu _0(G)=\frac {1}{|G|}\sum _{H\leq G,\, H\not \unlhd G}|H|$. We prove that $G$ is solvable if $\nu _0(G) \lt {29}/{6}$.