Question

How to find the order of an element in a cyclic group?

Answer 1

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The order of an elements g in a group G is the smallest number of times that you need to apply the group operation to g to obtain the identity.

Let G be cyclic of order 35. That means that there is an element g∈G with g35=e, and that gk≠e for all 1<k<35. Now, consider h=g5. Then h7=(g5)7=g35=e, but hk≠e for all 1<k<7, thus h has order 7. Similarly, the element g7 has order 5.

Remark: Cauchy's theorem states that if p is a prime dividing |G|, then G has an element of order p. Thus, the only finite groups where all elements except the identity have the same order are p-groups, namely groups whose order is a power of a fixed prime p. A group of size 35 is not a p-group.