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. \(p=(2{,}1,4{,}5,3{,}7,8{,}9,10{,}6)\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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