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Problem 60 (Properties Of $\langle a \rangle$ When $a$ Has Finite Order)

Let $G$ be a group with $a\in G$. Suppose that the order of $a$ is $|a|=n$. Prove the following:

  1. We have $\langle a\rangle = \{e,a,a^2,\ldots, a^{n-1}\}$. (You are showing two sets are equal.)
  2. We have $a^i=a^j$ if and only if $i-j$ is a multiple of $n$.
  3. The order of an element equals the order of the subgroup generated by that element, namely $|a|=|\langle a\rangle|$. (How can you combine 1 and 2 to get this.)

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