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 .
![{\displaystyle \mathbb {Z} [1/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6560a29f54d1403d584259891b958b24453fa1)
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 |