finiteorder

In mathematics, an element g of a group G has finite order if there exists a positive integer n such that g^n equals the identity element e. The least such n is called the order of g, denoted ord(g). Equivalently, the set {e, g, g^2, ..., g^{n-1}} forms a finite cyclic subgroup of G. If no such n exists, g has infinite order.

If G is finite, every element has finite order; in particular ord(g) divides the order of G by Lagrange’s theorem. The concept of finite order is central to the idea of torsion: the set of all finite-order elements in a group is called the torsion subset, and in an abelian group this set forms a subgroup known as the torsion subgroup.

Examples help illustrate the concept. In the symmetric group S_n, the order of a permutation equals the least common multiple of the lengths of its disjoint cycles. In a cyclic group of order m, every element has order that divides m, with generators attaining order m. In additive groups such as the integers, the only element of finite order is 0.

Finite order also appears in other contexts. For a linear operator on a vector space, having finite order means some positive power of the operator is the identity, which constrains the operator’s eigenvalues to be roots of unity and influences the operator’s diagonalizability in suitable bases.

Overall, the notion of finite order identifies elements that generate finite cyclic subgroups, providing a key tool for understanding the structure and behavior of groups across algebra and its applications.