I have recently been introduced to Sylow's theorem. I can't help but think how prime numbers are intimately related to group structure. For example, the fact that a power of a prime number being a divisor of the group's order necessitates the group have a subgroup of the same order is amazing and powerful.
What is it about prime numbers that makes it so? Is there something deeper that I am clearly missing about this apparent connection that prime numbers have with groups?