1 Definition and basic properties

The torsion subgroup of a group collects the elements that have finite order. In many settings it provides a compact way to separate the part of a group that eventually returns to the identity from the part that does not. The concept is especially natural in abelian group theory, where the finite-order elements always form a subgroup and therefore can be studied as an invariant in their own right.

1.1 Elements of finite order

An element \(g\) of a group has finite order if some positive integer \(n\) satisfies \(g^n = e\), where \(e\) is the identity element. The smallest such \(n\), when it exists, is called the order of \(g\). Elements of finite order are also called torsion elements. If no positive power of \(g\) equals the identity, then \(g\) has infinite order.

The set of torsion elements reflects how a group behaves under repeated multiplication. In cyclic notation, a finite-order element eventually cycles back to the identity, while an infinite-order element produces infinitely many distinct powers.

1.2 Torsion subgroup in abelian groups

For an abelian group, the collection of all torsion elements is closed under the group operation and inverses, so it forms a subgroup. This subgroup is called the torsion subgroup and is often denoted by \(T(G)\) or \(\operatorname{Tor}(G)\). It is a basic invariant of the group and is used to separate finite-order behavior from the torsion-free part.

1.2.1 Closure under the group operation

If \(a\) and \(b\) have finite order in an abelian group, then some positive integers \(m\) and \(n\) satisfy \(a^m=e\) and \(b^n=e\). Because the group is abelian, powers of \(ab\) can be rearranged conveniently, and one finds that \((ab)^{mn}=e\). Thus the product of two torsion elements is again torsion.

This closure can fail in non-abelian groups, where the order of multiplication matters. The abelian hypothesis is therefore essential for the torsion set to be a subgroup in general.

1.2.2 Identity and inverses

The identity element has order 1, so it is always torsion. If \(a\) has finite order, then \(a^{-1}\) also has finite order, since \((a^{-1})^n = (a^n)^{-1} = e\). These facts, together with closure under products in the abelian case, ensure that the torsion elements form a subgroup.

1.3 Torsion set in non-abelian groups

In a non-abelian group, the set of torsion elements need not be closed under multiplication. It may still contain many useful subgroups, but the full torsion set does not generally behave like a subgroup. This makes the non-abelian situation more delicate.

1.3.1 Failure to be a subgroup

A standard phenomenon in non-abelian groups is that two elements of finite order can have a product of infinite order. For this reason, the torsion elements of a non-abelian group are not always stable under the group law. The set can therefore fail the subgroup test.

1.3.2 Torsion elements and torsion subgroups

Even when the full torsion set is not a subgroup, a group may contain many torsion subgroups, meaning subgroups in which every element has finite order. These subgroups are studied individually and can reveal local or structural symmetry within the larger group. In some contexts, the largest torsion subgroup is of special interest, but it depends on the ambient group and may not coincide with the set of all torsion elements.

2 Examples

Examples clarify the range of behaviors that torsion can exhibit. Some groups are entirely torsion, while others contain both finite-order and infinite-order elements. The distinction is especially visible in familiar arithmetic and geometric groups.

2.1 Finite groups

Every element of a finite group has finite order, so a finite group is entirely torsion. In this case, the torsion subgroup is the whole group. This is one of the simplest settings in which the notion appears, although it becomes more informative in infinite groups.

2.2 Infinite abelian groups

Infinite abelian groups may have a nontrivial torsion subgroup, a torsion-free part, or both. Their examples show that finite-order elements can coexist with infinite-order ones in a highly organized way.

2.2.1 The integers

The additive group of integers \(\mathbb{Z}\) has no nonzero torsion elements. If \(n\mathbb{Z} = 0\) for some integer \(n>0\), then the integer must be 0. Thus \(\mathbb{Z}\) is torsion-free, and its torsion subgroup is trivial.

2.2.2 Direct sums of cyclic groups

A direct sum of finite cyclic groups is a typical torsion abelian group. For example, a group such as \(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}\) consists entirely of elements of finite order. More elaborate direct sums may include infinitely many summands, yet every element still has finite order because each element involves only finitely many coordinates.

2.3 Non-abelian examples

Non-abelian groups provide examples where torsion behaves less uniformly. Some are entirely torsion, while others mix finite and infinite orders in a way that prevents a simple subgroup description of all torsion elements.

2.3.1 Dihedral groups

Finite dihedral groups are torsion groups because they are finite. Infinite dihedral groups offer a more interesting case: they contain reflections of order 2, but also rotations that may have infinite order depending on the presentation. This makes them useful for illustrating the difference between finite-order elements and a torsion subgroup.

2.3.2 Matrix groups

Matrix groups often contain both torsion and non-torsion elements. For instance, a matrix with eigenvalues that are roots of unity may have finite order, while another with eigenvalues of infinite multiplicative order may not. Such examples are common in linear algebraic group theory and representation theory.

3 Structure in abelian groups

The torsion subgroup is one of the main structural tools for understanding abelian groups. It interacts naturally with decomposition theorems and often separates into more refined pieces according to prime divisibility.

3.1 Torsion part and torsion-free part

Every abelian group can be viewed as having a torsion part and a torsion-free part, though these parts do not always split as a direct sum in arbitrary groups. The torsion part consists of all finite-order elements, while the torsion-free part contains the elements whose nonzero multiples never vanish. This distinction is central in classification problems.

3.2 Maximal torsion subgroup

In an abelian group, the torsion subgroup is the unique maximal subgroup consisting entirely of torsion elements. Any torsion subgroup is contained in it. This maximality makes it a canonical subgroup, not dependent on auxiliary choices.

3.3 Primary decomposition

Torsion abelian groups often admit a decomposition into components associated with prime numbers. This organization reflects the way finite orders factor into prime powers and is a standard tool in the study of abelian groups.

3.3.1 p-torsion subgroup

For a prime \(p\), the \(p\)-torsion subgroup consists of elements whose order is a power of \(p\). Equivalently, it is the subgroup of elements annihilated by some power of \(p\). These subgroups isolate the \(p\)-primary behavior of the group.

3.3.2 Decomposition into p-primary components

A torsion abelian group can often be expressed as a direct sum of its \(p\)-primary components, one for each prime \(p\) that occurs in the orders of its elements. This decomposition organizes the group into simpler pieces and allows many questions to be reduced prime by prime.

3.4 Relation to finitely generated abelian groups

Finitely generated abelian groups have a particularly clean structure: they are isomorphic to a direct sum of a free abelian group and a finite torsion subgroup. In this classification, the torsion subgroup is the finite summand, and the free part is torsion-free. This decomposition is one of the foundational results of abelian group theory.

4 Torsion in modules

The idea of torsion extends naturally from groups to modules. In module theory, torsion measures whether a nonzero scalar can kill a vector-like element, and the resulting notions depend on the coefficient ring.

4.1 Torsion elements in modules over an integral domain

For a module over an integral domain, a torsion element is an element annihilated by some nonzero scalar from the ring. This generalizes the group-theoretic notion, since abelian groups may be regarded as modules over the integers. The relation to multiplication by integers makes the analogy especially close.

4.2 Torsion submodule

The torsion submodule of a module is the set of all torsion elements. Over an integral domain, this set is often a submodule and captures the part of the module that is killed by nonzero scalars.

4.2.1 Definition via annihilators

An element \(x\) of a module is torsion if there exists a nonzero ring element \(r\) such that \(rx=0\). The collection of all such annihilating scalars is called the annihilator of \(x\). Torsion theory examines how these annihilators vary across the module.

4.2.2 Examples and nonexamples

A module with only torsion elements is a torsion module, such as a finite abelian group viewed as a \(\mathbb{Z}\)-module. By contrast, a free module over an integral domain is torsion-free, since its basis elements cannot be killed by nonzero scalars. Mixed examples occur frequently and often require decomposition into torsion and torsion-free parts.

4.3 Torsion-free modules

A torsion-free module has no nonzero torsion elements. Such modules resemble vector spaces in the sense that nonzero scalars act injectively on their elements, though over a ring they may be far less rigid than vector spaces. Torsion-freeness is an important hypothesis in many algebraic constructions.

4.4 Torsion and localization

Localization often reduces or removes torsion phenomena by inverting selected nonzero elements of the ring. This can simplify a module by making some annihilators invertible, thereby separating local behavior from global structure. Torsion and localization are therefore closely linked in commutative algebra.

5 Applications

Torsion subgroups appear in many branches of algebra. They help identify finite-order structure, support classification theorems, and encode arithmetic information in a compact form.

5.1 Group classification problems

In the classification of abelian groups and related algebraic objects, torsion is one of the main invariants. Knowing the torsion subgroup can narrow the range of possible isomorphism types dramatically. It also interacts with generators, relations, and rank.

5.2 Algebraic number theory

Torsion concepts arise naturally in the arithmetic of rings and number fields. They help describe finite-order behavior among ideal-class and unit groups, where the distinction between finite and infinite components is especially significant.

5.2.1 Class groups and finite torsion

Class groups are finite in many important cases, so their elements are inherently torsion. This finiteness encodes arithmetic obstructions to unique factorization and plays a central role in algebraic number theory. Torsion language helps emphasize the finite-order nature of these groups.

5.2.2 Units and roots of unity

The group of units in a number ring often contains a finite torsion subgroup made up of roots of unity. This subgroup is typically small but important, since it represents the finite-order part of the multiplicative structure. The remaining units are usually organized by free abelian factors.

5.3 Homological algebra

Homological algebra frequently produces torsion groups and modules, especially in invariants built from chains and cycles. Torsion can reveal subtle structure that is invisible over fields.

5.3.1 Torsion in homology groups

Homology groups may contain torsion summands that detect finite-order phenomena in topological or algebraic objects. These torsion classes often carry information not visible in the free part of homology. As a result, torsion is a standard part of the interpretation of homological invariants.

5.3.2 Exact sequences and torsion behavior

Exact sequences are useful for tracking how torsion passes between related groups or modules. In many situations, torsion in one term influences torsion in neighboring terms, though the exact transfer depends on the maps involved. This makes torsion a natural companion to diagrammatic methods in homological algebra.

Several notions are closely tied to torsion and are frequently discussed alongside it. Together they describe the basic arithmetic of elements in groups and modules.

6.1 Torsion-free subgroup

A torsion-free subgroup is a subgroup in which every nonidentity element has infinite order. Such subgroups are important as complements or comparison objects for torsion subgroups. They often serve as the “continuous” or “unbounded” side of a decomposition.

6.2 Torsion group

A torsion group is a group in which every element has finite order. Finite groups are the most familiar examples, but infinite torsion groups also exist. The term emphasizes that torsion is not merely a subgroup phenomenon but can define the whole group.

6.3 Rank

In abelian group theory, rank measures the size of the largest free abelian part. It is closely related to torsion, since groups are often analyzed by separating a free component from a torsion component. Rank and torsion together provide a coarse but effective structural summary.

6.4 Order of an element

The order of an element is the smallest positive integer that sends it to the identity under repeated multiplication, when such an integer exists. It is the basic numerical measure underlying the notion of torsion. Finite order is precisely what makes an element torsion.