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    When a cyclic group is solvable

    P-TNPRX

    Area
    Algebra
    Topics
    Cyclic Groups, Solvable Groups
    Status
    draft

    Give a necessary and sufficient condition for a cyclic group to be solvable.

    Solution

    Every cyclic group is abelian, and every abelian group is solvable: its commutator subgroup is trivial.

    Thus if \(G\) is cyclic, \[ G'=[G,G]=1, \] so the derived series terminates after one step.

    Hence the necessary and sufficient condition is vacuous: \[ \boxed{\text{every cyclic group is solvable}.} \]

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