Sylow theorems

The Sylow theorems describe the existence, conjugacy, and number of subgroups whose orders are maximal powers of a prime dividing a finite group’s order.

Finite group: A group with finitely many elements, equipped with an associative operation, an identity, and inverses. The theorems concern prime-power subgroups inside groups with finitely many elements.
Sylow p-subgroup: A subgroup whose order is the largest power of p dividing the order of a finite group. This is the subgroup whose existence and behavior the theorems characterize.
Classification of groups of order p squared: The result that every group of order p squared is abelian, with two isomorphism types for each prime p. Sylow existence and normality arguments help reduce groups of prime-power order.
Sylow's first theorem: The theorem that a finite group has a subgroup whose order is the largest power of p dividing its order. It isolates the existence conclusion from the conjugacy and counting conclusions.
Subgroup: A subset of a group that is itself a group under the same operation. Each object classified by the theorems is a subgroup of the ambient finite group.
p-adic valuation: For a prime p, the exponent of p in an integer's prime factorization. It gives the exponent in the order of a Sylow p-subgroup.
Groups of order 12: The finite groups of order 12, including the cyclic group, the alternating group A4, and the dihedral group of order 12. Sylow counts constrain the numbers of subgroups of orders 3 and 4.
Sylow's second theorem: The theorem that every p-subgroup is contained in a Sylow p-subgroup and all Sylow p-subgroups are conjugate. It states the conjugacy and containment conclusions separately.
Lagrange's theorem: The order of a subgroup of a finite group divides the order of the group. It bounds possible subgroup orders and identifies the largest power of a chosen prime that can occur.
Normalizer: The subgroup of elements that conjugate a given subgroup to itself. The number of Sylow p-subgroups equals the index of their normalizer.

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