Exercise
Let binom(n,m) denote the binomial coefficient of n and m. Let n > 2 and let 1 < m < n be odd. The number of distinct m-cycles in An is given by
binom(n,m)(m - 1)!.
1/2 binom(n,m)(m - 1)!.
binom(n,m)m!.
This is true.
No. All odd m-cycles are even permutations.
No. Notice that (a1, a2, ... ,am), (a2, ... ,am,a1), ... ,(am, a1, ... ,am - 1) denote the same permutation.