Let be a group and a normal subgroup of . Prove that the order of the element in is , where is the least positive integer such that , and is infinite if no such exists. Give an example to show that the order of in may be strictly smaller than the order of in .
Suppose for all positive integers . Then we have for all positive integers , hence .
Suppose for some positive integer ; let be the least such integer. Then , so that . If is some integer strictly less than such that , then we have , a contradiction. Thus .
is normal in itself for all groups , and is the trivial group. So every element in has order 1, but elements of may have arbitrarily large order.