Number of Elements of order p in SpAn exercise from Herstein asks to prove that the number of elements of order p, p a ' in Sp, is (p−1)!+1. I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson's theorem which says (p−1)!≡−1 (mod p) using this result

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Number of Elements of order p in SpAn exercise from Herstein asks to prove that the number of elements of order p, p a &#039; in Sp, is (p−1)!+1. I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson&#039;s theorem which says (p−1)!≡−1 (mod p) using this result

Olive Guzman · Accepted Answer

Step 1Maybe you mean the number of elements of order dividing p (so that you are including the identity)? (Think about the case p=3 - there are two three cycles, not three of them.) For the general question, think about the possible cycle structure of an element of order p in Sp.You can go from the formula in your question to Wilson&#039;s theorem by counting the number of p-Sylow subgroups (each contains p−1 elements of order p), and then appealing to Sylow&#039;s theorem. (You will find that there are (p−2)! p-Sylow subgroups, and by Sylow&#039;s theorem this number is congruent to 1modp. Multiplying by p−1, we find that (p−1)! is congruent to −1modp.)

Aliana Porter · Accepted Answer

Every element of order p in Sp is a p-cycle. The symmetric group Sp−1 acts transitively on these p cycles.

Number of Elements of order p in S_{p} An exercise from Herstein asks to prove that the nu

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