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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.

While every nilpotent group is supersolvable, but the converse does not hold, as shown by these examples. Similarly, these examples show that not every supersolvable group is metacyclic.

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  • Review status: beta
  • Last edited by David Roe on 2021-10-08 14:24:56
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