Torsion-free group

From Groupprops
Jump to: navigation, search
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


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
VIEW: Definitions built on this | Facts about this: (facts closely related to Torsion-free group, all facts related to Torsion-free group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

Definition

A group is said to be torsion-free or aperiodic if it has no non-identity periodic element, or equivalently, if there is no non-identity element of finite order.

(The term aperiodic is sometimes also used with slightly different meanings, so torsion-free is the more unambiguous term).

Metaproperties

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 G is a torsion-free group and H is a subgroup of G. Then, H 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 G and a normal subgroup H of G such that the quotient group G/H is not a torsion-free group.
direct product-closed group property Yes torsion-freeness is direct product-closed If G_i, i \in I, are all torsion-free groups, so is the external direct product \prod_{i \in I} G_i.

Relation with other properties

Stronger properties

Prime-parametrized version