Group satisfying no nontrivial identity
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 in terms of words
A group satisfying no nontrivial identity is a group such that for any word with the property that:
we have that is a trivial word; in other words, if is a free group on generators , .
Definition in terms of verbal subgroups
A group satisfying no nontrivial identity is a group that cannot be expressed as the quotient of a free group by a normal subgroup that contains a nontrivial verbal subgroup.
Relation with other properties
Group satisfying a nontrivial identity: This is the precise opposite.