Group satisfying a nontrivial identity

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 $$G$$.

Opposite properties
Group satisfying no nontrivial identity is the precise opposite.

Stronger properties

 * Weaker than::Finite group: Here, the nontrivial identity is the fact that every element raised to a fixed finite power is the identity element.
 * Weaker than::Abelian group:
 * weaker than::Nilpotent group:
 * Weaker than::Solvable group:
 * Weaker than::Virtually Abelian group
 * Weaker than::Virtually nilpotent group
 * Weaker than::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 $$G/N$$.

A subgroup of a group satisfying a nontrivial identity also satisfies a nontrivial identity -- in fact, the same one.

A quotient group of a group satisfying a nontrivial identity also satisfies a nontrivial identity -- in fact, the same one.

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.