A Sylow p-subgroup is a maximal p-subgroup of a finite group, meaning it is a subgroup whose order is a power of a prime number p, and is not contained in any larger subgroup with that same property. These subgroups play a crucial role in understanding the structure of groups, especially when analyzing groups of small order and their classifications. The existence and number of Sylow p-subgroups are given by the Sylow theorems, which offer powerful tools for studying group properties and behaviors.
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