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    Nilpotent groups are solvable

    E-AMD-3TYWSS7F

    Area
    Algebra
    Topics
    Nilpotent Groups, Solvable Groups
    Status
    draft

    Show that \(G\) nilpotent \(\implies G\) solvable.

    Solution

    Let \(G\) be nilpotent. Then \(G\) has a finite central series \[ 1=Z_0\trianglelefteq Z_1\trianglelefteq\cdots\trianglelefteq Z_c=G \] with \[ Z_{i+1}/Z_i\le Z(G/Z_i) \] for every \(i\).

    Each quotient \(Z_{i+1}/Z_i\) is therefore abelian. Hence this central series is a finite subnormal series with abelian successive quotients, so \(G\) is solvable.

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