Discriminating group

Definition
A group $$G$$is termed a discriminating group if for any collection of words $$w_1,w_2,\dots,w_m$$ all in the letters $$x_1,x_2,\dots,x_n$$, the following are equivalent:


 * 1) For every $$(g_1,g_2,\dots,g_n) \in G^n$$, there exists $$i$$ with $$1 \le i \le m$$ such that $$w_i(g_1,g_2,\dots,g_n)$$ is the identity element of $$G$$.
 * 2) There exists $$i$$ with $$1 \le i \le m$$ such that for every $$(g_1,g_2,\dots,g_n) \in G^n$$, $$w_i(g_1,g_2,\dots,g_n)$$ is the identity element of $$G$$.

Another way of formulating this is that whenever a disjunction of words is satisfied in the group, one of the words must be satisfied in the group.

Facts

 * Discriminating and finite iff trivial: There is no nontrivial finite discriminating group.