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Exercise (Order Is The Smallest Positive Integer)
Suppose we know that $a^6=e$.
- Explain why this is not enough information to state the order of $a$. (Look at the definition. What are we missing?)
- In addition to knowing that $a^6=e$, someone else notices that $a^4=e$. Prove that the order of $a$ cannot be 4. In particular show that $a^2=e$, so the order of $a$ is either $2$ or $1$.
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- The order of an element is the SMALLEST positive integers $n$ such that $a^n=e$. If all we know is that $a^6=e$, then the order might be 6, or some number less.
- If we know that both $a^6=e$ and $a^4=e$, then since $6=4+2$ (the division algorithm), we know that $e=a^6=a^{4+2}=a^4a^2=ea^2=a^2$. This shows that $e=a^2$, which means the order now can at most be 2.