Sylow theorems

Sylow theorems are a set of results in group theory, a branch of mathematics, that describe the existence and arrangement of special subgroups within a finite group. These subgroups, called Sylow p-subgroups, are associated with prime numbers dividing the size of the entire group. The theorems guarantee that for each relevant prime, such subgroups exist, are related to each other in specific ways, and tell us about the structure and symmetry of the entire group. They are fundamental tools for understanding how complex groups are built from simpler, prime-based components.