The Sylow Theorems

The converse of Lagrange’s theorem is false: if G is a finite group and d|#G, then there may not be a subgroup of G with order d. The simplest example of this is the group A4, of order 12, which has no subgroup of order 6. The Norwegian mathematician Peter Ludwig Sylow [1] discovered that a converse result is true when d is a prime power: if p is a prime number and pk|#G then G must contain a subgroup of order pk. Sylow also discovered important relationships among the subgroups whose order is the largest power of p dividing #G, such as the fact that all subgroups of that order are conjugate to each other. For example, a group of order 100 = 22 · 52 must contain subgroups of order 1, 2, 4, 5, and 25, the subgroups of order 4 are conjugate to each other, and the subgroups of order 25 are conjugate to each other. It is not necessarily the case that the subgroups of order 2 are conjugate or that the subgroups of order 5 are conjugate.