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A group is supersolvable if there are nested normal subgroups $N_i\lhd G$

$${1}=N_0 < N_1 < N_2 < \ldots < N_k = G$$ with all successive quotients $N_{i+1}/N_i$ cyclic. It is stronger than requiring $N_{i+1}/N_i$ to be abelian (because $N_i \lhd G$), so not all solvable groups are supersolvable. The smallest solvable groups that are not supersolvable are $A_4$, $S_4$, SL$_2(\mathbb{F}_3)$ and $C_2\times A_4$. All supersolvable groups are monomial, and all abelian, nilpotent, metacyclic groups, Z-groups and groups of square-free order are supersolvable. Supersolvable groups are closed under subgroups and quotients (but not extensions), and a group is supersolvable if and only if each of its maximal subgroups has prime index.

Knowl status:
  • Review status: beta
  • Last edited by Meow Wolf on 2019-05-23 20:07:56
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