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 properties
VIEW 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