Characterization of the order of powers of a group element

Let G be a group. Suppose x \in G with |x| = n < \infty. If n = st for some positive integers s and t, prove that |x^s| = t.


We have (x^s)^t = x^{st} = x^n = 1, so that |x^s| \leq t. Now suppose that in fact |x^s| = u < t for some positive integer u. Then (x^s)^u = x^{su} = 1, where su < st = n. This gives a contradiction, so that no such u exists. Hence |x^s| = t.

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