LUCS-Baer Lie group

Definition
A group is termed a LUCS-Baer Lie group if it is a group of nilpotency class two (i.e., its derived subgroup is contained in ts center) and it satisfies the following equivalent conditions:


 * 1) Every element of its derived subgroup has a unique square root in the whole group.
 * 2) Every element of its derived subgroup has a unique square root among the elements in its center.
 * 3) Every element of the derived subgroup has a unique square root in the whole group and that square root is in the center.

Equivalence of definitions
Note that (3) implies both (1) and (2). The reverse implications are somewhat harder:


 * (1) implies (3): This follows from the fact that center is local powering-invariant. Note that we use class two in order to observe that the element whose square root we are taking is in the center to begin with.
 * (2) implies (3): This follows from the equivalence of definitions of nilpotent group that is torsion-free for a set of primes.