ABSTRACT
Lagrange’s Theorem 24.2 tells us that the order of every subgroup of a finite group must divide the order of the group. It is natural to ask if the converse of this theorem holds. We get a partial answer from Cauchy’s Theorem 24.5 that says that if the order of the group is divisible by prime p, then the group must have an element of order p. Thus the group must at least have a subgroup of order p. However, the converse of Lagrange’s Theorem does not hold in general, as we encountered in Example 24.6, where we discovered that the twelve element group A4 does not have a subgroup with 6 elements. (For alternative proofs for this result, see Exercise 24.19 or Exercise 27.10.)
Therefore, we cannot guarantee that in general there is a subgroup for every possible divisor of the order of the group. However, we can guarantee the existence of subgroups of some orders, find out the relationship between such subgroups, and even count how many such subgroups there are. These results are called the Sylow theorems (first proved by the Norwegian mathematician Peter Sylow, 1832-1918). These results focus attention on one prime divisor at a time.
