Equivalence of definitions of LUCS-Baer Lie group

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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

Statement

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

  1. Every element of the derived subgroup G' has a unique square root in G.
  2. Every element of its derived subgroup G' has a unique square root among the elements in the center Z(G).
  3. Every element of the derived subgroup G' has a unique square root in G and that square root is in the center Z(G).

Facts used

  1. Center is local powering-invariant
  2. 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 G 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 G, an element g \in G' such that there is a unique element x \in G satisfying x^2 = g.

To prove: x \in Z(G)

Proof: Since G has nilpotency class at most two, g \in G' \le Z(G), so g \in Z(G). The uniqueness of the square root, combined with Fact (1), tells us that x \in Z(G) as well.

(2) implies (3)

Given: A group G with the property that every element of G' has a unique square root in Z(G). An element g \in G' with square root x \in Z(G).

To prove: x is the unique square root of g in all of G.

Proof: Since every element of G' has a unique square root in Z(G), this in particular implies that the identity element has a unique square root in Z(G), so Z(G) is 2-torsion-free. By Fact (2) combined with G being nilpotent, this implies that the squaring map is injective from G to itself, so x is the unique square root of g in all of G.