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

Definition

A group $G$ is termed a group satisfying a nontrivial identity if there exists a nontrivial word $w(x_1,x_2,\dots,x_n)$ such that:

$w(g_1,g_2,\dots,g_n) = e \ \forall \ g_1, g_2, \dots, g_n \in G$.

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

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 $G$ has a normal subgroup $N$ such that both $N$ and $G/N$ satisfy (possibly different) nontrivial identities, so does $G$. In fact, we can write an identity satisfied by $G$ is expressible in terms of the identities known to be satisfied by $N$ and $G/N$. 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