Explanation:

To find the number of elements of order 6 in A6, we need to consider the cycle decomposition of each element of A6.

What is cycle decomposition?

Cycle decomposition is a way of representing a permutation as a product of disjoint cycles. A cycle is a sequence of elements that are permuted cyclically, i.e., each element is replaced by the next element in the cycle until we reach the first element again.

For example, the permutation (1 2 3)(4 5) can be decomposed into two cycles: (1 2 3) and (4 5).

What is order of a permutation?

The order of a permutation is the smallest positive integer k such that applying the permutation k times gives the identity permutation.

For example, the permutation (1 2 3)(4 5) has order 6, since applying it 6 times gives the identity permutation (1)(2)(3)(4)(5).

Why the answer is '0'?

Now, let's consider the elements of A6 of order 6. Since the order of a permutation equals the least common multiple of the lengths of its cycles, any element of order 6 must have a cycle decomposition consisting of cycles of lengths that multiply to 6.

However, since A6 consists only of even permutations, any cycle must have an even length. Therefore, the only possible cycle decompositions for an element of order 6 in A6 are:

- (1 2)(3 4)(5 6)
- (1 2 3)(4 5 6)

In the first case, the product of cycle lengths is 2 * 2 * 2 = 8, so this element cannot have order 6.

In the second case, the product of cycle lengths is 3 * 3 = 9, so this element also cannot have order 6.

Therefore, there are no elements of order 6 in A6.