Discriminating group
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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
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group is termed a discriminating group if for any collection of words all in the letters , the following are equivalent:
- For every , there exists with such that is the identity element of .
- There exists with such that for every , is the identity element of .
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)