how many subgroups?

Question

Let p be primes, how many subgroups of the order p does Sp have? what does np congruence 1 mod p say about (p-1)! and mod p?

ps: Sp is S sub p, and np is n sub p.

Answer 1

Any group of order p, for p a prime, is cyclic. So your question really asks how many elements of order p are there and how many of these are in the same subgroup? Well, using a cycle decomposition of the elements of the symmetric group, an element has order equal to the lcm of the sizes of the disjoint cycles. Since p is prime, this says that an element of order p is a p-cycle. How many *unequal* p-cycles are there? This is a question about how many ways you can order the numbers 1, 2, 3, ..., p, up to cyclic permutation, which you should be able to count easily. Now, how many p-cycles are in any cyclic group generated by a p-cycle? Count all these things and you'll have your answer. Sylow's Theorem is unnecessary.