What are laws characterizing the trivial group? I mean all group words $w$ such that if the identity $w=1$ holds in a group $G$, then $G=1$.

For example, it can be easily verified that if a group word $w=x_{i_1}^{\alpha_1}x_{i_2}^{\alpha_2}\cdots x_{i_t}^{\alpha_t}$ has the following property, then it characterizes the trivial group: $$\exists J\subseteq \{x_{i_1},\ldots, x_{i_t}\}:\ gcd(deg_{x_{i_j}}(w): x_{i_j}\in J)=1.$$

It seems that such a problem must be already studied. Anybody knows a reference?

P.S. Note that a given non-trivial group cannot be characterized by a law, because the identities of any group $A$ and its powers are the same.

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