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:

1. $G$ is a nilpotent group of nilpotency class at most $c$.
2. 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 $p\le i$.
   • 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 $p\le i$.
   • 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 $p\le i$, and this root is inside $Z^{c+1-i}(G)$.

Equivalence of definitions

Further information: equivalence of definitions of global LUCS-Lazard Lie group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
LUCS-Baer Lie group				|FULL LIST, MORE INFO
global Lazard Lie group				|FULL LIST, MORE INFO