2-powered group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group is termed a 2-powered group if it is powered over the prime 2, i.e., the square map is bijective from the group to itself, or equivalently, every element has a unique square root. Equivalently, the group is powered over the ring $\mathbb {Z} [1/2]$.

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction	Intermediate notions
odd-order group	finite group	|FULL LIST, MORE INFO
Baer Lie group	group of nilpotency class two	|FULL LIST, MORE INFO
2-powered nilpotent group	nilpotent group	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
2-torsion-free group	no element of order two			|FULL LIST, MORE INFO
2-divisible group	every element has a square root, i.e., the square map is surjective			|FULL LIST, MORE INFO
2-powering-injective group	the square map is injective			|FULL LIST, MORE INFO