Discriminating group

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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 $1\leq i\leq m$ such that $w_{i}(g_{1},g_{2},\dots ,g_{n})$ is the identity element of $G$.
2. There exists $i$ with $1\leq i\leq m$ 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.

References

• The verbal topology of a group by Roger M. Bryant, Journal of Algebra, ISSN 00218693, Volume 48,Number 2, Page 340 - 346(October 1977): ^{PDF (gated)More info}, Page 341 (Page 2 of the PDF)