Order (Group)

The number of elements in a Group $G$, denoted $\vert G\vert$. The order of an element $g$ of a finite group $G$ is the smallest Power of $n$ such that $g^n=I$, where $I$ is the Identity Element. In general, finding the order of the element of a group is at least as hard as factoring (Meijer 1996). However, the problem becomes significantly easier if $\vert G\vert$ and the factorization of $\vert G\vert$ are known. Under these circumstances, efficient Algorithms are known (Cohen 1993).

See also Abelian Group, Finite Group

References

Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993.

Meijer, A. R. ``Groups, Factoring, and Cryptography.'' Math. Mag. 69, 103-109, 1996.