Group satisfying no 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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Definition

Definition in terms of words

A group satisfying no nontrivial identity is a group such that for any word $w(x_{1},x_{2},\dots ,x_{n})$ with the property that:

$w(g_{1},g_{2},\dots ,g_{n})=e\ \forall g_{1},g_{2},\dots ,g_{n}\in G$,

we have that $w$ is a trivial word; in other words, if $F_{n}$ is a free group on $n$ generators $a_{1},a_{2},\dots ,a_{n}$, $w(a_{1},a_{2},\dots ,a_{n})=e$.

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.

Stronger properties

• Group having a free non-abelian subgroup