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    Lower and upper central series, nilpotent groups, and solvable groups

    P-FO36C

    Area
    Algebra
    Topics
    Nilpotent Groups, Solvable Groups, Subgroup Series
    Status
    draft

    Define lower central series, upper central series, nilpotent and solvable groups.

    Solution

    For a group \(G\), the lower central series is \[ \gamma_1(G)=G,\qquad \gamma_{i+1}(G)=[\gamma_i(G),G]. \] The upper central series is \[ Z_0(G)=1,\qquad Z_{i+1}(G)/Z_i(G)=Z\bigl(G/Z_i(G)\bigr). \]

    The group \(G\) is nilpotent if \(\gamma_{c+1}(G)=1\) for some \(c\ge0\); equivalently, \(Z_c(G)=G\) for some \(c\). The least such \(c\) is the nilpotency class.

    The derived series is \[ G^{(0)}=G,\qquad G^{(i+1)}=[G^{(i)},G^{(i)}]. \] The group \(G\) is solvable if \(G^{(r)}=1\) for some \(r\).

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