# Group satisfying no nontrivial identity

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

### Definition in terms of words

A **group satisfying no nontrivial identity** is a group such that for any word with the property that:

,

we have that is a *trivial* word; in other words, if is a free group on generators , .

### Definition in terms of verbal subgroups

A **group satisfying no nontrivial identity** is a group that cannot be expressed as the quotient of a free group by a normal subgroup that contains a nontrivial verbal subgroup.

## Relation with other properties

### Opposite properties

Group satisfying a nontrivial identity: This is the precise opposite.