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