2-powered group
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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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Contents
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 .
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 |