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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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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(The term aperiodic is sometimes also used with slightly different meanings, so torsion-free is the more unambiguous term).
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|quasivarietal group property||Yes||torsion-freeness is quasivarietal||The condition of being torsion-free can be described by a collection of quasi-identities.|
|subgroup-closed group property||Yes||torsion-freeness is subgroup-closed||Suppose is a torsion-free group and is a subgroup of . Then, is also a torsion-free group.|
|quotient-closed group property||No||torsion-freeness is not quotient-closed||It is possible to have a torsion-free group and a normal subgroup of such that the quotient group is not a torsion-free group.|
|direct product-closed group property||Yes||torsion-freeness is direct product-closed||If , are all torsion-free groups, so is the external direct product .|
Relation with other properties
- Torsion-free group for a set of primes: Given a set of primes , a -torsion-free group is a group that has no element of order for any .