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

.

Therefore

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 .

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