Definition
In mathematics, a torsion group (also called a periodic group) is a group in which every element has finite order; that is, for each element $g$ there exists a positive integer $n$ such that $g^{n}=e$, where $e$ denotes the identity element of the group.
Overview
Torsion groups arise naturally in various branches of algebra, particularly in the study of abelian groups, group cohomology, and the classification of finite groups. The concept contrasts with that of a torsion‑free group, where no non‑identity element has finite order. Important examples include finite groups (every element has bounded order) and the additive group of rational numbers whose denominators are powers of a fixed prime. In the abelian setting, the structure theorem for finitely generated abelian groups decomposes any such group into a direct sum of a free (torsion‑free) part and a torsion subgroup.
Etymology/Origin
The term “torsion” is borrowed from the notion of torsion elements in group theory, itself derived from the Latin torsio meaning “twist” or “turn.” Historically, the terminology emerged in the late 19th and early 20th centuries as algebraists formalised concepts of element order and periodicity within groups.
Characteristics
- Element order: For every $g \in G$ there exists $n \in \mathbb{N}$ such that $g^{n}=e$.
- Exponent: The exponent of a torsion group is the least common multiple of the orders of all its elements (if such a finite integer exists); groups with a finite exponent are called bounded torsion groups.
- Subgroup structure: Any subgroup of a torsion group is itself a torsion group. Conversely, a quotient of a torsion group may fail to be torsion if the normal subgroup contains elements of infinite order (though this cannot occur when the original group is torsion).
- Abelian torsion groups: In the abelian case, torsion groups decompose into direct sums of p‑primary components, each consisting of elements whose orders are powers of a fixed prime $p$.
- Relation to finiteness: All finite groups are torsion, but infinite torsion groups also exist; classic examples include the Prüfer $p$-group $\mathbb{Z}(p^\infty)$ and certain Burnside groups constructed via the Burnside problem.
Related Topics
- Torsion element – an individual element of finite order within a group.
- Torsion‑free group – a group in which the only element of finite order is the identity.
- Burnside problem – a set of questions concerning the existence and properties of finitely generated groups of bounded exponent (i.e., bounded torsion groups).
- Periodic group – another term for a torsion group, emphasizing the periodic nature of element orders.
- p‑group – a group where the order of every element is a power of a fixed prime $p$; a special class of torsion groups.
- Structure theorem for finitely generated abelian groups – a foundational result describing how such groups split into torsion and free components.