The permutation σ ∈ S2020 has its order = 2019. What is the maximum number of fixed points?
I was thinking that since its order is 2019, the permutation with 2019 different elements is equal to the neutral permutation. So in essence, there is only one point which stays the same, and by that I mean fixed. Is this correct or am I missing something ?
$$2019=3×673$$ We can achieve a permutation of order $2019$ by letting there be a $3$-cycle and a $673$-cycle. Thus $2020-3-673=1344$ points are fixed, and since there is no other factorisation of $2019$ with smaller sum of factors, this is the best we can do.