The number of elements in a group , denoted . If the order of a group is a finite number, the group is said to be a finite group.
The order of an element of a finite group is the smallest power of such that , where 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 and the factorization of are known. Under these circumstances, efficient algorithms are known (Cohen 1993).
The group order can be computed in the Wolfram Language using the function GroupOrder[n].
More things to try:
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.
