All possible types of permutation.

Question

A permutation $ \sigma \in S_{10} $ satisfies the conditions $$ \forall_{1 \le i \le 29} \sigma^i \neq id, \sigma^{30} = id $$ Determine all possible types of the permutations. Give me a hand.

Answer 1

The "type" of a permutation means its cycle type, i.e., how it factors into disjoint cycles. For example, here is a permutation of type $3,3,2,1,1$: $$\sigma=(1\ 2\ 3)(4\ 5\ 6)(7\ 8)(9)(10)$$ This is not the one of the permutations you're looking for because $\sigma^6=\mathrm{Id}$.