Global LUCS-Lazard Lie group

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Definition

A group is termed a global LUCS-Lazard Lie group if there exists a positive integer $c$ such that:

$G$ is a nilpotent group of nilpotency class at most $c$ .
The following equivalent formulations:
- For every positive integer $i$ , every element of the $i^{th}$ lower central series member $\gamma _{i}(G)$ has a unique $p^{th}$ root in $G$ for all $p\leq i$ .
- For every positive integer $i$ , every element of the $i^{th}$ lower central series member $\gamma _{i}(G)$ has a unique $p^{th}$ root in the upper central series member $Z^{c+1-i}(G)$ for all $p\leq i$ .
- For every positive integer $i$ , every element of the $i^{th}$ lower central series member $\gamma _{i}(G)$ has unique $p^{th}$ roots in $G$ for all $p\leq i$ , and this root is inside $Z^{c+1-i}(G)$ .