Discriminating group

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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

A group Gis 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

References