Sylow's Theorems

Sylow's theorems are a set of results in group theory that describe how certain subgroups, called Sylow p-subgroups, exist within a finite group. These subgroups are associated with prime number factors of the group's size. The theorems guarantee the existence of these subgroups, detail their possible number, and explain how they relate to one another through conjugation—meaning they can be transformed into each other via group elements. Overall, Sylow's theorems help mathematicians understand the internal structure of groups by revealing how these prime-specific building blocks are organized and interconnected.