# Group satisfying a nontrivial identity

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 **group satisfying a nontrivial identity** if there exists a *nontrivial* word such that:

.

In other words, there is a nontrivial identity that is satisfied universally in .

## Relation with other properties

### Opposite properties

Group satisfying no nontrivial identity is the precise opposite.

### Stronger properties

- Finite group: Here, the nontrivial identity is the fact that every element raised to a fixed finite power is the identity element.
`For full proof, refer: Finite implies nontrivial identity` - Abelian group:
`For full proof, refer: Abelian implies nontrivial identity` - Nilpotent group:
`For full proof, refer: Nilpotent implies nontrivial identity` - Solvable group:
`For full proof, refer: Solvable implies nontrivial identity` - Virtually Abelian group
- Virtually nilpotent group
- Virtually solvable group

## Metaproperties

If a group has a normal subgroup such that both and satisfy (possibly different) nontrivial identities, so does . In fact, we can write an identity satisfied by is expressible in terms of the identities known to be satisfied by and . `For full proof, refer: Satisfying a nontrivial identity is extension-closed`

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property

View a complete list of subgroup-closed group properties

A subgroup of a group satisfying a nontrivial identity also satisfies a nontrivial identity -- in fact, the same one. `For full proof, refer: Satisfying a nontrivial identity is subgroup-closed`

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

A quotient group of a group satisfying a nontrivial identity also satisfies a nontrivial identity -- in fact, the same one. `For full proof, refer: Satisfying a nontrivial identity is quotient-closed`

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property

View other finite direct product-closed group properties

A direct product of finitely many groups satisfying nontrivial identities also satisfies a nontrivial identity. This is, in fact, a special case of the result mentioned above for extensions. `For full proof, refer: Satisfying a nontrivial identity is finite-direct product-closed`