# Reduced free 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

No. | Shorthand | A group is termed a reduced free group if ... | A group is termed a reduced free group if ... |
---|---|---|---|

1 | Quotient of free | it is isomorphic to the quotient of a free group by a verbal subgroup | there is a free group and a verbal subgroup of such that |

2 | Free in some subvariety | it is a free algebra in some subvariety of the variety of groups. | there is a subvariety of the variety of groups such that is a free algebra in that subvariety. More explicitly, there is a generating set for such that for any , any set map extends uniquely to a group homomorphism from to . |

3 | Free in own subvariety | it is a free algebra in the subvariety of the variety of groups generated by itself. | is a free algebra in the subvariety (i.e., the subvariety generated by ) in the variety of groups. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Free group | free in the variety of groups | |FULL LIST, MORE INFO | ||

Free abelian group | free in the variety of abelian groups | |FULL LIST, MORE INFO | ||

Elementary abelian group | trivial or abelian of prime exponent | |FULL LIST, MORE INFO | ||

Burnside group | free in the variety of groups of exponent dividing , for some pre-specified | |FULL LIST, MORE INFO | ||

Finite homocyclic group | finite direct power of a finite cyclic group | |FULL LIST, MORE INFO | ||

Free class two group | free in the variety of groups of nilpotency class two | |FULL LIST, MORE INFO | ||

Free metabelian group | free in the variety of metabelian groups | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Group in which every fully invariant subgroup is verbal | every fully invariant subgroup is a verbal subgroup | fully invariant implies verbal in reduced free | simple groups give counterexamples | |FULL LIST, MORE INFO |

## Metaproperties

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