## Powers without the identity

### Task number: 2462

Find a permutation on 10 elements s.t. $$p^i$$ is not the identity (i.e. $$p^i\ne\imath$$) for all $$i=1,…,29$$.

• #### Resolution

If the permutation $$p$$ has cycles of length $$d_1,…,d_k$$ we get the identity in the power $$p^i$$, where the exponent $$i$$ divides all $$d_1,…,d_k$$. On 10 elements we can take any permutation of cycle lengths 2, 3 a 5, e.g. $$p=(2{,}1,4{,}5,3{,}7,8{,}9,10{,}6)$$.