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## Main Question or Discussion Point

Here is the statement: If x is an element of finite order ##n## in ##G##, prove that the elements ##1,x,x^2, \dots , x^{n-1}## are all distinct.

So I started by reinterpreting the question into something more tangible. Is it true that this problem is equivalent to the following? Suppose that ##|x| = n## and that ##0 \le k,m \le n-1##. Prove that ##x^k = x^m ## if and only if ##k = m##.

So I started by reinterpreting the question into something more tangible. Is it true that this problem is equivalent to the following? Suppose that ##|x| = n## and that ##0 \le k,m \le n-1##. Prove that ##x^k = x^m ## if and only if ##k = m##.