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2-powered group

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 2-powered nilpotent group|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