2-powered group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group]] is termed a '''2-powered group''' if it is [[powered group for a set of primes|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. | A [[group]] is termed a '''2-powered group''' if it is [[defining ingredient::powered group for a set of primes|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 [[defining ingredient::group powered over a unital ring|powered over]] the ring <math>\mathbb{Z}[1/2]</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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| [[Weaker than::Baer Lie group]] || [[group of nilpotency class two]] || {{intermediate notions short|2-powered group|Baer Lie group}} | | [[Weaker than::Baer Lie group]] || [[group of nilpotency class two]] || {{intermediate notions short|2-powered group|Baer Lie group}} | ||
|- | |||
| [[Weaker than::2-powered nilpotent group]] || [[nilpotent group]] || {{intermediate notions short|2-powered group|2-powered nilpotent group}} | |||
|} | |||
===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::2-torsion-free group]] || no element of order two || || || {{intermediate notions short|2-torsion-free group|2-powered group}} | |||
|- | |||
| [[Stronger than::2-divisible group]] || every element has a square root, i.e., the square map is surjective || || || {{intermediate notions short|2-divisible group|2-powered group}} | |||
|- | |||
| [[Stronger than::2-powering-injective group]] || the square map is injective || || || {{intermediate notions short|2-powering-injective group|2-powered group}} | |||
|} | |} | ||
Latest revision as of 16:13, 5 August 2013
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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 |