6.3 The Sylow theorems and their applications

The Sylow theorems are game-changers in group theory. They tell us about special subgroups called Sylow p-subgroups in finite groups. These theorems help us understand group structure and are key to classifying finite groups.

Knowing the Sylow theorems is crucial for tackling problems in group theory. They're like a Swiss Army knife for group analysis, helping us figure out subgroup existence, count subgroups, and even prove some groups aren't simple. Super handy stuff!

Sylow theorems for finite groups

Introduction to Sylow theorems

• The Sylow theorems are a set of three powerful theorems in group theory that provide information about the existence and properties of certain subgroups of a finite group, known as Sylow p-subgroups
• These theorems are named after the Norwegian mathematician Ludwig Sylow, who proved them in 1872
• The Sylow theorems are essential tools in the study of finite groups, as they allow for the classification of groups of small order and the determination of various structural properties of finite groups
• The theorems are particularly useful in the study of finite simple groups, which are the building blocks of all finite groups

Statements of the Sylow theorems

• The first Sylow theorem states that if G is a finite group and p is a prime number dividing the order of G, then G contains a Sylow p-subgroup, which is a subgroup of order p^k, where p^k is the highest power of p dividing the order of G
• The second Sylow theorem states that all Sylow p-subgroups of a finite group G are conjugate to each other, meaning that if P and Q are Sylow p-subgroups of G, then there exists an element g in G such that $gPg^{-1} = Q$
• The third Sylow theorem states that the number of Sylow p-subgroups of a finite group G, denoted by n_p, satisfies the following conditions:
  • n_p divides the order of G
  • n_p is congruent to 1 modulo p
  • $n_p = [G : N_G(P)]$, where $N_G(P)$ is the normalizer of a Sylow p-subgroup P in G

Existence and number of Sylow p-subgroups

Determining the existence of Sylow p-subgroups

• To determine the existence of a Sylow p-subgroup in a finite group G, one needs to check if the order of G is divisible by a prime number p
• If the order of G is divisible by p, then by the first Sylow theorem, G contains at least one Sylow p-subgroup
• For example, if G is a group of order 60, then G has Sylow 2-subgroups (order 4), Sylow 3-subgroups (order 3), and Sylow 5-subgroups (order 5)

Calculating the number of Sylow p-subgroups

• The number of Sylow p-subgroups, denoted by n_p, can be determined using the third Sylow theorem
• n_p must divide the order of G and be congruent to 1 modulo p
• The number of Sylow p-subgroups can also be calculated using the formula $n_p = [G : N_G(P)]$, where $N_G(P)$ is the normalizer of a Sylow p-subgroup P in G
• In some cases, the number of Sylow p-subgroups can be uniquely determined based on the order of the group and the prime number p
  • For example, if p does not divide $[G : N_G(P)]$, then there is only one Sylow p-subgroup
• If a group G has a unique Sylow p-subgroup for some prime p, then this subgroup must be normal in G

Classifying groups of small order

Groups of prime power order

• For groups of order p^n, where p is a prime number, there is only one Sylow p-subgroup, which is the group itself
• This implies that groups of prime power order are always solvable
• Examples of groups of prime power order include the cyclic group of order p (Z_p) and the dihedral group of order 2^n (D_{2^n})

Groups of order pq

• For groups of order pq, where p and q are distinct primes, the Sylow theorems can be used to show that such groups are either cyclic or isomorphic to a semidirect product of cyclic groups
• If p does not divide q-1, then the group must be cyclic (Z_pq)
• If p divides q-1, then the group is isomorphic to a semidirect product of Z_q and Z_p (Z_q ⋊ Z_p)
• Examples of groups of order pq include Z_15 (cyclic) and Z_3 ⋊ Z_5 (non-cyclic)

Proving the non-simplicity of certain groups

• The Sylow theorems can be used to prove that certain groups of small order, such as groups of order 15 or 21, are not simple and must have normal subgroups
• For a group of order 15, the Sylow theorems imply that there is only one Sylow 3-subgroup and one Sylow 5-subgroup, both of which must be normal
• For a group of order 21, the Sylow theorems imply that there is only one Sylow 7-subgroup, which must be normal

Structure and properties of finite groups

Existence of normal subgroups

• The Sylow theorems can be used to prove the existence of normal subgroups in certain finite groups
• If a group G has a unique Sylow p-subgroup for some prime p, then this subgroup must be normal in G
• The existence of normal subgroups is essential in the study of group extensions and the classification of finite groups

Solvability of certain groups

• The Sylow theorems can be used to prove the solvability of certain groups, such as groups of prime power order or groups with a unique Sylow p-subgroup for some prime p
• A group G is solvable if it has a series of subgroups $1 = G_0 ⊴ G_1 ⊴ ... ⊴ G_n = G$, where each quotient $G_{i+1} / G_i$ is abelian
• The solvability of a group has important implications for its structure and representation theory

Orders of elements in finite groups

• The Sylow theorems can be used to determine the possible orders of elements in a finite group
• If G is a finite group and p is a prime dividing the order of G, then G must contain elements of order p
• The existence of elements of certain orders can provide information about the structure and properties of the group, such as its center or its conjugacy classes

Applications to finite simple groups

• The Sylow theorems are particularly useful in the study of finite simple groups, as they provide information about the possible orders and structures of such groups, which can be used to classify them or prove their uniqueness
• For example, the Sylow theorems can be used to prove that the alternating group A_5 is the only non-abelian simple group of order 60
• The classification of finite simple groups, which was completed in the early 1980s, relies heavily on the Sylow theorems and their applications to the structure and properties of finite groups