Let n greaterthanorequalto 2 and sigma subset Sn be a cycle

Let n greaterthanorequalto 2 and sigma subset S_n be a cycle. Show that sigma is a k-cycle if and only if |sigma| = k.

Solution

we are given that sigma is a cycle of permutation group of size n. by definition if it has k elements then it is k cycle.To prove that if has k elements then it is K-cycle. that is trivial too because if it is not k-cycle then it is less than k-cycle but then elements in sigma are also less than k which is a contradiction, hence only if part.