A group is nilpotent exactly when its lower central series reaches the identity.Suggest a slogan
Let \(G\) be a group with identity \(e\) and lower central series \(G^0\geq G^1\geq\cdots\). Then \(G\) is nilpotent if and only if \(G^n=\theset{e}\) for some \(n\geq0\).
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T-7PU33Nilpotent groups via the upper central seriesvariant-of