# Discriminating group

## Contents

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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.