Question

How to find howmany elements of orde 2 in a group order n?

Answer 1

I know how to find the order of element in any group n, for example the order of 22 in Z5Z5is 55 as 2+2+2+2+2=[10]5=002+2+2+2+2=[10]5=00, which is the identity in Z5Z5.

But, how to calculate number of element of particular order in symmetric group SnSn?

I know how to calculate order of an element in SnSn, for example in S5S5, (123)(45)(123)(45) has an order 66. But how to calculate easily number of such elements? That is, how to calculate number of elements order 55, number of elements of order 44 in S5S5? Is there is any easy formula? Also is there is any formula to calculate the number of elements of particular order in the alternating group AnAnas well?