Equivalence of definitions of LUCS-Baer Lie group
This article gives a proof/explanation of the equivalence of multiple definitions for the term LUCS-Baer Lie group
View a complete list of pages giving proofs of equivalence of definitions
- Every element of the derived subgroup has a unique square root in .
- Every element of its derived subgroup has a unique square root among the elements in the center .
- Every element of the derived subgroup has a unique square root in and that square root is in the center .
- Center is local powering-invariant
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
(3) implies (1)
This is immediate.
(3) implies (2)
If the square root is unique in all of and also happens to be in the center, it must be unique among the elements of the center as well.
(1) implies (3)
Given: A group , an element such that there is a unique element satisfying .
Proof: Since has nilpotency class at most two, , so . The uniqueness of the square root, combined with Fact (1), tells us that as well.
(2) implies (3)
Given: A group with the property that every element of has a unique square root in . An element with square root .
To prove: is the unique square root of in all of .
Proof: Since every element of has a unique square root in , this in particular implies that the identity element has a unique square root in , so is 2-torsion-free. By Fact (2) combined with being nilpotent, this implies that the squaring map is injective from to itself, so is the unique square root of in all of .