# 2-powered group

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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 |