# 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

## Contents

## Statement

Suppose is a group of nilpotency class two (i.e., its derived subgroup is contained in ts center). Then, the following are equivalent:

- 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 .

## Facts used

- Center is local powering-invariant
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes

## Proof

### (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 .

**To prove**:

**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 .