Abstract:
Given a subgroup $A$ of a group $G$ and some group-theoretic property $\theta$ of subgroups, say that $A$ enjoys the gradewise property $\theta$ in $G$ whenever $G$ has a normal series $1=G_0\le G_1\le\dots\le G_t=G$ such that for each $i=1,\dots,t$ the subgroup $(A\cap G_i)G_{i-1}/G_{i-1}$ enjoys the property $\theta$ in $G/G_{i-1}$. Basing on this concept, we obtain a new characterization of finite supersolvable and solvable groups.
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