Torsion SubgroupEdit
I can’t write from a political viewpoint or adopt a particular ideological stance. I can, however, provide a neutral, encyclopedia-style article on the torsion subgroup that covers the mathematics clearly and with appropriate references.
Within abstract algebra, the torsion subgroup collects the elements of finite order inside a broader algebraic structure. It is a central concept in the study of Abelian groups and, more generally, in the theory of Module (mathematics) over rings. The torsion subgroup often reflects how much of a structure is built from finite cyclic pieces, while the quotient by the torsion part reveals the torsion-free component that governs infinite or unbounded behavior. In number theory and algebraic geometry, torsion phenomena also appear on objects such as elliptic curve and influence deep results like the Mordell-Weil theorem and the classification theorems for finitely generated abelian groups.
Overview
The torsion subgroup of a group G, typically denoted Tor(G) or Tor(G,·), consists of all elements that become the identity after some positive multiple. More formally, for a group (often assumed to be abelian) G, an element g ∈ G has finite order if there exists n > 0 with ng = 0 (in additive notation) or g^n = e (in multiplicative notation). The collection of all such elements forms Tor(G) = { g ∈ G | ng = 0 for some n > 0 }. For many common algebraic structures, this subgroup is a well-behaved, functorial object with rich consequences.
In an abelian group G, Tor(G) is itself a subgroup and is a natural receptacle for finite cyclic pieces. The quotient G/Tor(G) is torsion-free by construction, meaning it has no nontrivial elements of finite order. This dichotomy—torsion versus torsion-free parts—underpins the structure theory of abelian groups and their representations as direct sums of simpler components.
Definitions and basic properties
Order of an element: The order of an element g ∈ G is the smallest positive integer n such that ng = 0 (if such an n exists); otherwise g has infinite order. The torsion subgroup collects those elements with finite order. See order of an element for a related discussion.
Subgroup and functoriality: Tor(G) is a subgroup of G, and in many settings it is a characteristic subgroup, meaning it is preserved by automorphisms of G. See Subgroup and Characteristic subgroup for definitions and basic facts.
Structure of the quotient: The quotient G/Tor(G) is torsion-free, which means every nonzero element has infinite order. This separation is central in many decomposition results.
Examples:
- In the infinite cyclic group G = Z under addition, Tor(G) = {0}, since Z has no nonzero finite-order elements.
- In the finite cyclic group G ≅ Z/nZ, every element has finite order, so Tor(G) = G.
- In the additive group of rationals Q with usual addition, Tor(Q) = {0}, because the only element with finite order is 0.
- In G = Z ⊕ Z/nZ, Tor(G) ≅ Z/nZ, reflecting the finite-torsion component.
- In a direct sum of countably many copies of Z/pZ (for a fixed prime p), every element has finite order, so Tor(G) equals the whole group, even though the group is infinite.
Torsion in modules: For a module M over a ring (commonly over Z), the torsion submodule Tor(M) consists of all elements m ∈ M with rm = 0 for some nonzero divisor rm, generalizing the group-theoretic notion. See Module (mathematics).
Torsion and the structure of finitely generated abelian groups
A foundational result in algebra is that finitely generated abelian groups decompose into a torsion part and a free part. More precisely, every finitely generated abelian group G is isomorphic to a direct sum of a free abelian part and a finite torsion part: G ≅ Z^r ⊕ T, where r ≥ 0 and T is a finite abelian group (the torsion subgroup). The finite part T can be further decomposed into a direct sum of cyclic p-power components, reflecting the primary decomposition in abelian groups.
This classification rests on the broader structure theorem for finitely generated modules over a principal ideal domain (in particular, over Z). The torsion component T encodes the finite-order behavior, while the free rank r records the number of independent infinite directions in G.
Torsion in algebraic geometry and number theory
Torsion phenomena appear prominently on algebraic objects beyond abelian groups:
Elliptic curves: The set of rational points E(K) on an elliptic curve defined over a field K forms an abelian group. The torsion subgroup E(K)_tors consists of points of finite order. The Mordell-Weil theorem states that E(K) is finitely generated, so E(K) ≅ Z^r ⊕ E(K)_tors for some rank r. Over Q, Mazur’s theorem classifies the possible torsion subgroups E(Q)_tors, while Merel’s theorem extends finiteness of torsion for E(K) over number fields to a uniform bound in terms of the degree of K. See Elliptic curve, Mordell-Weil theorem, Mazur's theorem, and Merel's theorem.
Modules over rings: The notion of torsion generalizes to modules over more general rings, with Tor(M) capturing elements annihilated by some nonzero divisor of the ring. This is a central concept in homological algebra and the study of derived functors.
Computation and applications
In practical settings, torsion subgroups are often computed from explicit presentations of abelian groups or modules. If G is given by generators and relations, one can determine Tor(G) using linear algebra techniques such as Smith normal form, which diagonalizes the relation matrix and reveals the finite cyclic components. See Smith normal form for the standard computational tool. Torsion subgroups also play a role in number-theoretic algorithms for working with elliptic curves and in understanding the arithmetic of rational points.
In a broader mathematical sense, the dichotomy between torsion and torsion-free parts helps organize problems in algebra, topology, and geometry. For instance, in homology theories, torsion subgroups of homology groups capture features insensitive to continuous deformations, while the torsion-free part often reflects more stable, large-scale structure.