If $G$ is a group, for subsets $A$ and $B$ let $[A,B]$ be the subgroup generated by commutators of $A$ and $B$, i.e.,
\[ [A,B] = \langle a^{-1}b^{-1}ab\mid a\in A \text{ and } b\in B\rangle.\]
Further, let $G^0=G$ and for $i>0$, $G^i = [G,G^{i-1}]$. Then, the **lower central series** of $G$ is a normal series of subgroups
\[ G \unrhd G^1 \unrhd G^2 \unrhd \cdots .\]
Then, $G$ is nilpotent if and only if $G^k =\langle e\rangle$ for some $k>0$.

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- Review status: reviewed
- Last edited by John Jones on 2019-06-12 14:38:52

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- 2019-06-12 14:38:52 by John Jones (Reviewed)