Find the number of elements of order 5 and 6 in 𝐴_6

The alternating group A_6 is the group of all even permutations of 6 elements.

1. Elements of order 5 in A_6:

The elements of order 5 in A_6 are 5-cycles. We can form a 5-cycle from 6 elements in 6P5 = 720 ways. However, each 5-cycle can be written in 5 different ways (by cyclically permuting the elements), so we have over-counted by a factor of 5. Therefore, the number of elements of order 5 in A_6 is 720/5 = 144.

2. Elements of order 6 in A_6:

The elements of order 6 in A_6 are the product of a 2-cycle and a 4-cycle. We can choose the 2 elements for the 2-cycle in 6C2 = 15 ways, and then the remaining 4 elements can be arranged in a 4-cycle in (4-1)! = 6 ways. However, each such product can be written in 2 different ways (by swapping the 2-cycle and the 4-cycle), so we have over-counted by a factor of 2. Therefore, the number of elements of order 6 in A_6 is 15*6/2 = 45.

So, there are 144 elements of order 5 and 45 elements of order 6 in A_6.