decide $\sigma^{2020}$

Question

I don't know how to do this. Please help me.

Answer 1

Apply the permutation three times to any integer between 1 and 10 and observe you obtain the same integer back; i.e. $\sigma^3$ is the identity permutation. This means we can perform $\sigma^3$ an arbitrary number of times and it has no overall effect.

Since $2020=673\cdot 3+1$, we observe that we can write ${\sigma_3}^{2020}$ as ${\sigma_3}\circ ({\sigma_3}^3)^{673}$. What this is saying is that we effectively 'do nothing' 673 times, represented by the $({\sigma_3}^3)^{673}$, then apply once more the original permutation, to make a total of 2020 applications of the permutation.

So in fact ${\sigma_3}^{2020}={\sigma_3}\circ ({\sigma_3}^3)^{673}={\sigma_3}$.

I hope this helps.