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
Examples help illustrate the concept. In the symmetric group S_n, the order of a permutation equals the
Finite order also appears in other contexts. For a linear operator on a vector space, having finite
Overall, the notion of finite order identifies elements that generate finite cyclic subgroups, providing a key