In the last post we counted the number of elements of order 2 in the symmetric group . In exactly the same way we can count the number of elements of order , where is a prime number. Let this number be denoted by . Then
Here we adopt the convention if . Note that
by Wilson’s theorem. So
If and are base expansions, then using Lucas’ theorem one has
This generalizes the result from the previous post.
Corollary. The number of subgroups of order in is congruent to .
Proof. If there are subgroups of order in , then and so .