Sylow theorems
The Sylow theorems are three fundamental results in finite group theory that address the existence, conjugacy classes, and enumeration of maximal subgroups of prime power order, known as Sylow p-subgroups, within a finite group G of order p^k m where p is prime and does not divide m.[1] These theorems, first proved by Norwegian mathematician Peter Ludvig Sylow in his 1872 paper "Théorèmes sur les groupes de substitutions," provide essential insights into the internal structure of finite groups by guaranteeing the presence of such subgroups and relating their properties to the group's order.[2] Sylow's First Theorem asserts that for every prime p dividing the order of G, there exists at least one Sylow p-subgroup of order p^k, and every p-subgroup of G is contained in some Sylow p-subgroup.[3] Sylow's Second Theorem states that all Sylow p-subgroups of G are conjugate to each other, meaning that for any two such subgroups P and Q, there exists an element g in G such that Q = g P g^{-1}.[1] Sylow's Third Theorem specifies that the number n_p of Sylow p-subgroups divides m and satisfies n_p \equiv 1 \pmod{p}, which often implies n_p = 1 (and thus a normal Sylow p-subgroup) under certain conditions on the group order.[3] Beyond their foundational role in proving the existence and equivalence of these subgroups, the Sylow theorems enable powerful applications across mathematics. In finite group theory, they facilitate the classification of groups of small orders, such as showing that every group of order 15 is cyclic by establishing unique normal Sylow subgroups of orders 3 and 5.[4] They also underpin the proof that the alternating group A_5 is the smallest non-abelian simple group, by analyzing Sylow subgroups to rule out normal subgroups in groups of orders up to 60.[4] In arithmetic, the theorems contribute to Wilson's theorem, confirming that a natural number p is prime if (p-1)! \equiv -1 \pmod{p} via the action of the symmetric group S_p on its Sylow p-subgroups.[4] Overall, these theorems remain indispensable for decomposing finite groups into their p-primary components and advancing the broader classification of finite simple groups.[1]Fundamentals
Historical Context and Motivation
The Sylow theorems emerged in the mid-19th century as a pivotal advancement in finite group theory, building on foundational results by earlier mathematicians. Peter Ludvig Mejdell Sylow (1832–1918), a Norwegian mathematician and educator, formalized these theorems in a seminal 10-page paper published in 1872.[5] This work extended Augustin-Louis Cauchy's 1845 theorem, which established the existence of subgroups of prime order p in any finite group whose order is divisible by p, and drew inspiration from Évariste Galois's 1830s investigations into permutation groups and the solvability of polynomial equations by radicals.[6] Sylow's contributions were motivated by the broader quest to classify finite groups by dissecting their structure according to the prime factors of their orders, particularly focusing on the role of p-power subgroups in revealing underlying symmetries.[7] The motivation for the Sylow theorems stemmed from the limitations of Lagrange's theorem, which states that the order of any subgroup divides the group's order but provides no guarantee of the existence of subgroups for divisors beyond primes. Cauchy and Galois had highlighted the importance of prime-order subgroups in permutation representations and equation solvability, prompting deeper inquiry into higher prime powers. Sylow, through his study of Galois's unpublished manuscripts around 1870, recognized the need for a converse-like result: if a prime p divides the group order to the power k, then maximal subgroups of order p^k must exist to capture the full p-primary component of the group's structure. This approach facilitated the decomposition of finite groups into their Sylow p-subgroups, aiding classification efforts by isolating "p-parts" akin to primary decomposition in abelian groups.[8][7] Intuitively, the existence of maximal p-subgroups arises from the idea that groups with p dividing their order must incorporate elements or cycles whose orders are powers of p, ensuring a largest such subgroup to "saturate" the p-factor. Consider the symmetric group S_3 of order 6 = 2 × 3, where the Sylow 2-subgroup is any subgroup of order 2 generated by a transposition (e.g., \langle (1\ 2) \rangle), and the Sylow 3-subgroup is the alternating subgroup A_3 of order 3 generated by a 3-cycle (e.g., \langle (1\ 2\ 3) \rangle). These maximal p-subgroups reflect the group's permutation action, where p-cycles or products force the presence of p-power structure without exceeding the available order. In larger symmetric groups like S_p, the Sylow p-subgroup consists of permutations fixing all but p points, illustrating how the full p-power divides the order and demands a dedicated maximal subgroup to account for the p-sylow symmetries. Sylow's 1872 paper, titled "Théorèmes sur les groupes de substitutions," was published in French in the German journal Mathematische Annalen, reflecting the international mathematical community's lingua franca at the time despite Sylow's Norwegian background. Earlier, Sylow had delivered lectures on Galois theory in Norwegian at the University of Christiania (now Oslo) in 1862–1863, introducing advanced substitution group concepts to a local audience, but the theorems themselves appeared first in this accessible European outlet. The paper's publication accelerated the development of abstract group theory in the late 19th century, influencing figures like Sophus Lie and Felix Klein, though its Norwegian origins and Sylow's limited subsequent output somewhat delayed widespread adoption until translations and citations proliferated in the 1880s and beyond.[5]Precise Statement of the Theorems
A Sylow p-subgroup of a finite group G, where p is a prime and |G| = p^k m with p \nmid m, is defined as a subgroup of G of order p^k.[2] Sylow's first theorem asserts that for every prime p dividing |G|, G possesses at least one Sylow p-subgroup.[2] Sylow's second theorem states that any two Sylow p-subgroups of G are conjugate in G.[2] Sylow's third theorem declares that if n_p denotes the number of Sylow p-subgroups of G, then n_p \equiv 1 \pmod{p} and n_p divides m; moreover, n_p = 1 if and only if the Sylow p-subgroup is unique and hence normal in G.[2]Properties and Consequences
Existence and Uniqueness Conditions
The first Sylow theorem establishes the existence of Sylow p-subgroups in any finite group G. For a prime p dividing the order |G| of G, let p^k denote the highest power of p dividing |G|; then G contains at least one subgroup P of order p^k, and any such subgroup is termed a Sylow p-subgroup of G. The order of every Sylow p-subgroup is thus uniquely fixed as p^k, independent of the specific choice of subgroup, providing a canonical measure of the p-primary component in the prime factorization of |G|. This existence result, originally proved by Peter Ludvig Sylow in 1872, forms the foundation for analyzing the internal structure of finite groups by isolating their maximal p-subgroups. A Sylow p-subgroup P of G is unique if and only if the number n_p of distinct Sylow p-subgroups satisfies n_p = 1. In this case, P is characteristic in G and hence normal, meaning gPg^{-1} = P for all g \in G. When P is normal, the Schur--Zassenhaus theorem guarantees the existence of a p-complement H, a subgroup of order |G|/p^k such that G = PH and P \cap H = \{e\}, with G isomorphic to the semidirect product P \rtimes H. This decomposition highlights the structural interplay between the Sylow p-subgroup and the complementary Hall subgroup, enabling recursive analysis of G by reducing to smaller orders coprime to p. The existence of Sylow p-subgroups facilitates a decomposition of G into its p-parts across distinct primes, as the order of G factors into products of such p^k terms, allowing the group to be viewed through the lens of its primary components. p-Complements exist more generally under the hypotheses of the Schur--Zassenhaus theorem, which applies whenever a normal Sylow p-subgroup meets a coprime-order complement, but their existence without normality requires additional conditions like solvability. Uniqueness fails precisely when n_p > 1, as occurs in non-abelian simple groups, where no nontrivial proper subgroup is normal, ensuring multiple conjugate Sylow p-subgroups for each p dividing |G|.Conjugacy and Counting Formula
A fundamental aspect of the Sylow theorems concerns the conjugacy of Sylow p-subgroups in a finite group G. Specifically, any two Sylow p-subgroups P and Q of G are conjugate, meaning there exists an element g \in G such that Q = gPg^{-1}. This conjugacy implies that all Sylow p-subgroups of G are isomorphic to one another.[2] The number n_p of distinct Sylow p-subgroups of G, where |G| = p^k m with p \nmid m, satisfies the conditions n_p \equiv 1 \pmod{p} and n_p divides m. These constraints arise from index considerations in the structure of G relative to the Sylow p-subgroups.[1] This counting formula is closely tied to the natural action of G on the set \mathrm{Syl}_p(G) of its Sylow p-subgroups by conjugation, where the orbit-stabilizer theorem yields n_p = [G : N_G(P)] for any Sylow p-subgroup P, with N_G(P) denoting the normalizer of P in G. The kernel of this action is contained in every normalizer N_G(Q) for Q \in \mathrm{Syl}_p(G).[9]Immediate Corollaries
A fundamental immediate corollary of the Sylow theorems concerns groups whose order is a power of a prime. If the order of a finite group G is p^k for some prime p and integer k \geq 1, then by the existence theorem, G possesses a Sylow p-subgroup P of order p^k. Since |G:P| = 1, the counting theorem implies that the number of Sylow p-subgroups n_p divides 1, so n_p = 1. Thus, P = G is the unique Sylow p-subgroup of G, and it is normal in G.[10] Furthermore, every nontrivial finite p-group has a nontrivial center, as established by the class equation applied to p-groups, where the center Z(G) must contain a non-identity element to account for the p-power order. Another key consequence arises when all Sylow subgroups of G are normal, meaning n_p = 1 for every prime p dividing |G|. In this case, the Sylow p-subgroups P_p for distinct primes p pairwise intersect trivially, since their orders are powers of different primes. Moreover, their product \prod P_p equals G by Lagrange's theorem, as the order multiplies to |G|. Since each P_p is normal in G, the subgroups centralize one another (as [P_p, P_q] \leq P_p \cap P_q = \{e\} for p \neq q), so G is isomorphic to the direct product of its Sylow subgroups. The converse also holds: if G is the direct product of its Sylow subgroups, then each is normal in G. This characterization applies in general, though such groups are nilpotent and hence solvable.[10] The Sylow theorems also yield information about the distribution of elements of prime order. For a prime p dividing |G|, fix a Sylow p-subgroup P. Each of the remaining n_p - 1 Sylow p-subgroups intersects P in a proper subgroup (by the conjugacy theorem and nontriviality), so they each contain at least p-1 elements of order p outside P. Thus, the total number of elements of order p in G is at least (n_p - 1)(p - 1). This lower bound limits the possible values of n_p, as the p-elements (including the identity and those of higher p-power order) occupy a significant portion of G, thereby constraining the number of non-p-elements and aiding in the analysis of group orders.[10] When n_p = 1 for every prime p dividing |G|, the above direct product structure implies that G possesses a normal p-complement for each p (namely, the direct product of the other Sylow q-subgroups for q ≠ p).Examples and Applications
Basic Examples in Finite Groups
The symmetric group S_3 has order 6, which factors as $2 \times 3.[11] Its Sylow 2-subgroups are the cyclic subgroups of order 2 generated by transpositions, such as \langle (1\,2) \rangle = \{e, (1\,2)\}, \langle (1\,3) \rangle = \{e, (1\,3)\}, and \langle (2\,3) \rangle = \{e, (2\,3)\}; there are three such subgroups, so n_2 = 3.[11] The Sylow 3-subgroup is unique, \langle (1\,2\,3) \rangle = \{e, (1\,2\,3), (1\,3\,2)\}, with n_3 = 1, making it normal in S_3.[11] To verify conjugacy of the Sylow 2-subgroups, explicit computation shows that conjugation by elements of S_3 permutes them: for instance, conjugating \langle (1\,2) \rangle by (1\,3) yields \langle (2\,3) \rangle, since (1\,3)(1\,2)(1\,3)^{-1} = (2\,3), and similarly for the others.[1] The number n_2 = 3 follows from Sylow's third theorem, as it divides 3 and is congruent to 1 modulo 2.[1] For the Sylow 3-subgroup, n_3 = 1 divides 2 and is congruent to 1 modulo 3, confirming uniqueness.[1] The subgroup lattice of S_3 consists of the trivial subgroup at the bottom, connected to the three Sylow 2-subgroups and the single Sylow 3-subgroup in the middle layer, all of which are maximal and connect directly to S_3 at the top; this structure highlights the normal Sylow 3-subgroup as the only one containing no proper nontrivial subgroups beyond the trivial one.[12] The alternating group A_4 has order 12, which factors as $2^2 \times 3.[1] Its Sylow 2-subgroup is unique, the Klein four-group V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}, with n_2 = 1, making it normal in A_4.[1] The Sylow 3-subgroups are cyclic of order 3, such as \langle (1\,2\,3) \rangle = \{e, (1\,2\,3), (1\,3\,2)\}, \langle (1\,2\,4) \rangle, \langle (1\,3\,4) \rangle, and \langle (2\,3\,4) \rangle; there are four such subgroups, so n_3 = 4.[13] Conjugacy among the Sylow 3-subgroups holds by Sylow's second theorem; for example, conjugation by (3\,4) maps \langle (1\,2\,3) \rangle to \langle (1\,2\,4) \rangle, since (3\,4)(1\,2\,3)(3\,4)^{-1} = (1\,2\,4).[13] The number n_3 = 4 satisfies Sylow's third theorem, dividing 4 and congruent to 1 modulo 3, while n_2 = 1 divides 3 and is congruent to 1 modulo 2.[1] The subgroup lattice of A_4 features the trivial subgroup at the base, rising to the four Sylow 3-subgroups and the unique Sylow 2-subgroup V (which itself has three subgroups of order 2), with these connecting to intermediate structures like the normalizer of V before reaching A_4 at the apex; this illustrates how the normal Sylow 2-subgroup serves as a key building block in the lattice.[14]Applications to Group Structure and Classification
The Sylow theorems provide powerful tools for analyzing the structure of finite groups of small order, particularly in establishing non-simplicity. For a finite group G with |G| \leq 60, the constraints imposed by Sylow's third theorem on the number n_p of Sylow p-subgroups—namely, n_p divides |G|/p^k where p^k is the highest power of p dividing |G|, and n_p \equiv 1 \pmod{p}—combined with the assumption of simplicity (no nontrivial normal subgroups, so n_p > 1 for all p) lead to contradictions for all such orders except 60. Specifically, for each order up to 60, either some n_p = 1 (implying a normal Sylow subgroup) or the action of G on the set of Sylow p-subgroups yields a nontrivial homomorphism to a symmetric group whose order does not divide |G|, violating simplicity. This analysis confirms that no non-abelian simple groups exist below order 60, with the alternating group A_5 of order 60 serving as the smallest example, where n_2 = 5, n_3 = 10, and n_5 = 6 satisfy the conditions without contradiction.[4] In the classification of groups of order pq where p < q are distinct primes, the Sylow theorems determine the possible structures precisely. The number n_q of Sylow q-subgroups divides p and satisfies n_q \equiv 1 \pmod{q}; since p < q, the only possibility is n_q = 1, making the Sylow q-subgroup normal in G. For n_p, it divides q (so n_p = 1 or q) and n_p \equiv 1 \pmod{p}. If n_p = 1, both Sylow subgroups are normal, and G is cyclic (isomorphic to \mathbb{Z}_{pq}). If n_p = q, this requires q \equiv 1 \pmod{p} for consistency, yielding a non-abelian semidirect product \mathbb{Z}_q \rtimes \mathbb{Z}_p. For instance, the group of order 15 (p=3, q=5) has $5 \not\equiv 1 \pmod{3}, so n_3 = 1 and G \cong \mathbb{Z}_{15}. In contrast, order 21 (p=3, q=7) allows n_3 = 7 since $7 \equiv 1 \pmod{3}, producing both the cyclic \mathbb{Z}_{21} and a non-abelian group where \mathbb{Z}_3 acts on \mathbb{Z}_7 via an automorphism of order 3 (e.g., multiplication by 2 modulo 7).[10] Sylow p-subgroups control the fusion of p-elements in G, meaning two p-elements are conjugate in G if and only if they are conjugate in the normalizer of some Sylow p-subgroup; this localization of conjugacy classes aids in dissecting the group's structure via its p-local properties. A related structural implication arises when n_p \leq p for every prime p dividing |G|: since n_p \equiv 1 \pmod{p}, the possible values force n_p = 1 (as the next candidate $1 + p > p), rendering all Sylow subgroups normal and G the direct product of its Sylow p-subgroups, which is solvable. This criterion highlights how bounded Sylow numbers enforce solvability through normalized decomposition.[1] Computational classifications of finite groups up to order $10^6 leverage Sylow theorems to constrain isomorphism types by computing possible n_p values, enumerating Sylow subgroups, and constructing extensions or semidirect products systematically. Algorithms in systems like GAP implement Sylow-based backtracking to generate all groups, for example, confirming the complete enumeration up to order $10^6 while ensuring no overlooked structures via exhaustive Sylow intersection checks. These efforts build on earlier manual classifications for smaller orders, extending them efficiently.[15]Connections to Number Theory and Other Theorems
The Sylow theorems find significant applications in number theory, notably in proofs of Wilson's theorem, which states that for a prime p, (p-1)! \equiv -1 \pmod{p}.[16] One such proof utilizes the structure of the symmetric group S_p, whose order is p!. The Sylow p-subgroups of S_p are cyclic of order p, generated by p-cycles, and their number n_p satisfies n_p \equiv 1 \pmod{p} by Sylow's third theorem and divides (p-1)!. Counting the p-cycles shows there are (p-1)! such elements, each Sylow p-subgroup contains p-1 non-identity elements, and distinct Sylow p-subgroups intersect trivially, yielding n_p = (p-2)!. Thus, (p-1)! = (p-1) \cdot n_p \equiv (p-1) \cdot 1 = -1 \pmod{p}, confirming Wilson's theorem.[17] This approach highlights how the counting formula from Sylow's theorems ties directly to factorial congruences modulo primes. In the multiplicative group (\mathbb{Z}/p\mathbb{Z})^*, which has order p-1, the Sylow q-subgroups for primes q dividing p-1 provide insight into the group's cyclic structure and the existence of primitive roots. Since (\mathbb{Z}/p\mathbb{Z})^* is cyclic, each Sylow q-subgroup is unique (n_q = 1) and itself cyclic. For a finite abelian group, the condition that all Sylow subgroups are cyclic implies the group is cyclic, as it decomposes as a direct product of these cyclic Sylow subgroups of coprime orders.[18] Primitive roots modulo p are the generators of (\mathbb{Z}/p\mathbb{Z})^*, and their classification relies on this Sylow decomposition: an element generates the full group if and only if its projection generates each Sylow q-subgroup to full order. The number of such primitive roots is \phi(p-1), reflecting the Euler totient function applied to the order. This connection underscores how Sylow theorems elucidate the subgroup lattice essential for primitive root properties in modular arithmetic.[19] More broadly, Sylow subgroups of the general linear group \mathrm{GL}_n(\mathbb{F}_p) link group theory to modular representation theory and aspects of class field theory. In \mathrm{GL}_n(\mathbb{F}_p), the Sylow p-subgroups are the unipotent upper triangular matrices with 1s on the diagonal, of order p^{n(n-1)/2}, and their conjugates describe the p-local structure relevant to representations over fields of characteristic p.[10] These structures appear in the study of modular representations of finite groups, where restriction to Sylow p-subgroups helps decompose irreducible representations and compute decomposition numbers. In class field theory, particularly through Artin reciprocity, Galois representations into \mathrm{GL}_n over \mathbb{Q}_p or finite fields involve p-Sylow subgroups to analyze ramification and inertia groups in abelian extensions, connecting local reciprocity laws to global number-theoretic invariants.[20] A related observation ties back to factorial congruences: in the symmetric group S_{p-1}, whose order (p-1)! is not divisible by p, there is no non-trivial Sylow p-subgroup, hence none that is normal. This absence ensures that all elements of order p in S_p lie outside the point stabilizers isomorphic to S_{p-1}, facilitating the exact counting of such elements via Sylow subgroups and reinforcing the modular arithmetic of factorials in Wilson's theorem.[4]Proofs
Proof of Existence
The existence of Sylow p-subgroups for a finite group G with |G| = p^k m where p \nmid m is established constructively by starting with a nontrivial p-subgroup and iteratively enlarging it until a subgroup of order p^k is obtained.[1] By Cauchy's theorem, G contains an element of order p, so there exists a p-subgroup H \leq G of order p. Suppose |H| = p^l with l < k, so p divides the index [G : H] = p^{k-l} m. To extend H, consider the set \Omega of left cosets G/H, with |\Omega| = [G : H]. The subgroup H acts on \Omega by left multiplication: for h \in H and gH \in \Omega, define h \cdot (gH) = (hg)H.[1] A coset gH \in \Omega is fixed by this action if h(gH) = gH for all h \in H, which is equivalent to g^{-1}Hg \leq H. Since |g^{-1}Hg| = |H|, it follows that g^{-1}Hg = H, so g \in N_G(H), the normalizer of H in G. Thus, the fixed cosets are precisely those in N_G(H)/H, and the number of fixed points is | \mathrm{Fix}_H(\Omega) | = [N_G(H) : H].[1] Since H is a p-group acting on \Omega, the orbit-stabilizer theorem implies that the size of each orbit divides |H| = p^l, so is a power of p. Therefore, the number of fixed points (orbits of size 1) satisfies |\Omega| \equiv |\mathrm{Fix}_H(\Omega)| \pmod{p}, or equivalently, [G : H] \equiv [N_G(H) : H] \pmod{p}. [1] As p divides [G : H], it follows that p divides [N_G(H) : H]. Note that H < N_G(H) \leq G, so N_G(H)/H is a nontrivial group of order divisible by p. By Cauchy's theorem applied to N_G(H)/H, there exists a subgroup L/H \leq N_G(H)/H of order p. The preimage L = \pi^{-1}(L/H), where \pi : N_G(H) \to N_G(H)/H is the quotient map, is then a p-subgroup of G containing H with |L| = p \cdot |H| = p^{l+1}.[1] Replacing H by L and repeating the process yields a chain of p-subgroups with strictly increasing orders. Since |G| is finite, this process terminates after finitely many steps, producing a p-subgroup P \leq G such that p \nmid [G : P]. By Lagrange's theorem, |P| = p^k, so P is a Sylow p-subgroup of G.[1]Proofs of Conjugacy and Counting
The conjugacy of Sylow p-subgroups, known as Sylow's second theorem, asserts that for a finite group G and prime p, any two Sylow p-subgroups P and Q satisfy Q = gPg^{-1} for some g \in G. [1] To prove this, consider the action of Q on the set of left cosets G/P by left multiplication. [1] Since |G/P| = |G|/|P| = m with p \nmid m, the fixed-point congruence for p-group actions implies that the number of fixed points is congruent to m modulo p, hence nonzero. [1] Thus, there exists a coset gP fixed by every element of Q, meaning qgP = gP for all q \in Q, or equivalently, Q \leq gPg^{-1}. [1] As |Q| = |gPg^{-1}| = p^k, it follows that Q = gPg^{-1}. [1] This fixed-point congruence arises from the class equation applied to the action: for a p-group Q acting on a set X, the average number of fixed points over elements of Q equals |X|/|Q|, but non-identity elements fix a number of points divisible by p (since their orbits have size a power of p), so |X| \equiv |\mathrm{Fix}(Q)| \pmod{p}. [21] Sylow's third theorem, the counting theorem, states that if |G| = p^k m with p \nmid m, then the number n_p of Sylow p-subgroups satisfies n_p \equiv 1 \pmod{p} and n_p \mid m. [1] Let \mathrm{Syl}_p(G) denote this set. The group G acts on \mathrm{Syl}_p(G) by conjugation, and by the conjugacy theorem, this action is transitive, forming a single orbit of size n_p. [1] By the orbit-stabilizer theorem, n_p = |G : N_G(P)| for any Sylow p-subgroup P, where N_G(P) is the normalizer of P in G. [1] Since P \trianglelefteq N_G(P), |P| = p^k divides |N_G(P)|, so n_p = |G| / |N_G(P)| divides |G| / p^k = m. [21] To establish n_p \equiv 1 \pmod{p}, consider the action of P on \mathrm{Syl}_p(G) by conjugation. [1] The fixed points of this action are the Q \in \mathrm{Syl}_p(G) such that pQp^{-1} = Q for all p \in P, i.e., P \leq N_G(Q). [1] For such a Q, both P and Q are Sylow p-subgroups of N_G(Q), and |N_G(Q)| = |G|/n_p is divisible by p^k since n_p \mid m and p \nmid m. [21] Moreover, Q \trianglelefteq N_G(Q) by definition of the normalizer. [1] In N_G(Q), the conjugacy theorem implies all Sylow p-subgroups are conjugate, but since Q is normal, its conjugates in N_G(Q) are Q itself, making Q the unique Sylow p-subgroup of N_G(Q). [1] Thus, P = Q, so P is the only fixed point. [1] By the fixed-point congruence for the p-group P acting on \mathrm{Syl}_p(G), n_p \equiv 1 \pmod{p}. [1] A variant of Burnside's lemma underlies this congruence: the number of orbits is the average number of fixed points, but for p-groups, it yields the modulo p relation directly via the above reasoning. [21] Consequently, |N_G(P)| = p^k \cdot (m / n_p), where m / n_p divides m. [1]Generalizations
Sylow Theorems for Infinite Groups
In the context of infinite groups, particularly profinite groups, the notion of a Sylow p-subgroup is adapted to the topological setting, where it is defined as a closed pro-p subgroup P of G that is maximal among closed pro-p subgroups, or equivalently, such that the quotient G/P is a profinite group whose order (in the sense of supernatural numbers) is coprime to p. This definition generalizes the finite case by emphasizing p-local properties and maximality with respect to the pro-p topology, ensuring P captures the p-primary component of G. For profinite groups, analogs of the Sylow theorems hold via p-adic completions and the structure of inverse limits. Every profinite group G possesses Sylow p-subgroups for each prime p, which are precisely the closed pro-p subgroups maximal with respect to inclusion among closed pro-p subgroups; their existence follows from the fact that any closed pro-p subgroup is contained in a maximal one, often constructed as the inverse limit of Sylow p-subgroups in the finite quotients of G. Moreover, all such Sylow p-subgroups are conjugate within G.[22] In compact profinite groups, conjugacy of Sylow p-subgroups extends further when G is p-complete, meaning G coincides with its p-adic completion; under this condition, any two Sylow p-subgroups are conjugate by an element of G, preserving the pro-p structure. This property is crucial in settings where the topology enforces completeness, such as in the study of absolute Galois groups. Modern applications of these generalized Sylow theorems appear prominently in algebraic number theory, particularly for infinite Galois groups like the absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), which is profinite. Here, the Sylow p-subgroups describe the maximal pro-p extensions unramified outside finitely many primes, enabling analysis of p-adic phenomena; for odd primes \ell, these subgroups decompose as semidirect products F \rtimes \mathbb{Z}_\ell where F is a free pro-\ell group on countably infinitely many generators, facilitating computations in Iwasawa theory and class field theory.[22]Extensions to p-Solvable and Fusion Contexts
In p-solvable finite groups, the Sylow theorems extend with additional structural control, particularly bounding the number n_p of Sylow p-subgroups in terms of the group's p-length, which measures the complexity of its p-composition series. Specifically, for a p-solvable group G expressed as a product of p-solvable subgroups A and B with additional conditions like p-nilpotency in one factor, the p-length \ell_p(G) satisfies \ell_p(G) \leq \max\{1 + \tau_p(A)/2, 1 + \tau_p(B)/2\}, where \tau_p relates to the Sylow numbers influencing fusion and permutability; this implies n_p is constrained by exponential bounds tied to the p-solvability index, preventing unbounded growth seen in arbitrary finite groups. Alperin's fusion theorem further refines this by asserting p-local control: for a Sylow p-subgroup P of G, the fusion of subsets A, B \subseteq P (where B = A^g for g \in G) is realized through a product of p-elements in normalizers of tame Sylow intersections P \cap Q_i (with Q_i other Sylow p-subgroups) and an element in N_G(P), ensuring that the normalizer N_G(P) dictates p-element conjugation within P.[23][24] This fusion mechanism implies key consequences for transfer homomorphisms, as the focal subgroup P \cap G' of P is generated by commutators [P, N_G(P)] and those from tame intersections, linking local p-structure to global properties like the existence of normal p-complements in p-solvable contexts. In fully solvable groups—a special case of p-solvable for all primes—the Sylow p-subgroups are complemented by Hall subgroups of p'-order, guaranteeing a subgroup K such that G = P \rtimes K with |K| coprime to p. Gaschütz's theorem strengthens this for normal abelian subgroups N \trianglelefteq G: if N has a complement in some subgroup H with \gcd(|N|, |G:H|) = 1, then N complements in G, applicable to Sylow settings where abelian Sylow p-subgroups in solvable G admit such splits; extensions show complements exist if all Sylow subgroups of N are abelian, relying on Šemetkov's criterion for coprime actions.[24][25][26] Post-2000 advancements, leveraging the Classification of Finite Simple Groups (CFSG), have clarified Sylow fusion in simple groups via saturated fusion systems, abstracting the normalizer action on Sylow p-subgroups. For instance, all reduced saturated fusion systems on p-groups of nilpotency class two are classified, yielding a new proof of Gilman and Gorenstein's theorem that finite simple groups with such Sylow 2-subgroups are restricted to specific families like alternating or sporadic types, with fusion controlled by N_G(P)/C_G(P); this refines Alperin's results by confirming realizability in simple groups without exotic fusions beyond CFSG bounds. These developments underscore p-local methods in simple group structure, where fusion patterns distinguish quasisimple extensions.[27]Computational Methods
Algorithms for Computing Sylow Subgroups
The computation of Sylow p-subgroups in finite groups given by generators is essential for structural analysis in computational group theory. For permutation groups, the process typically begins by applying the Schreier-Sims algorithm to obtain a base and strong generating set, which provides an efficient framework for subgroup membership tests and orbit computations. This structure enables a constructive backtrack search to build the Sylow p-subgroup incrementally.[28] The basic algorithm starts with a trivial subgroup or one generated by an element of maximal p-power order found via random selection or systematic search in the group. It then extends this initial p-subgroup H by identifying elements g in the group such that g normalizes H and the order of \langle H, g \rangle is strictly larger than |H| by a factor of p^m for some m \geq 1. The backtrack search leverages the stabilizer chain from the Schreier-Sims representation: at each stabilizer level, representatives of p-orbits are examined to find suitable extensions that preserve the chain while increasing the p-part of the order. The process repeats, branching on possible choices and pruning paths where the projected order cannot reach the full p^k (the p-part of the group order, computed from the BSGS), until a subgroup of order p^k is obtained. By Sylow's existence theorem, this maximal p-subgroup is a Sylow p-subgroup.[28][29] To confirm the result, the normalizer N_G(P) of the constructed Sylow p-subgroup P is computed iteratively using the stabilizer chain and backtrack on cosets of P. This involves finding the largest subgroup containing P that acts by conjugation on P, verified by checking that the index [G : N_G(P)] \equiv 1 \pmod{p} and divides the p'-part of |G|, consistent with Sylow's counting theorem.[28] In matrix groups over finite fields, specialized methods exploit the representation. When the field has characteristic p, the Sylow p-subgroup is unipotent; it can be constructed by applying p-modular reduction (if starting from characteristic zero or mixed characteristic) followed by computing row echelon forms of matrices to identify a basis in which the subgroup acts as upper-triangular unipotent matrices with ones on the diagonal. This involves iteratively finding invariant flags via Gaussian elimination on random elements, building the unipotent radical step-by-step until the order matches p^k. For fields of characteristic not p, the Sylow p-subgroup is semisimple; the construction uses diagonalization over extensions or finding commuting semisimple elements of p-power order, often reducing to the permutation case by acting on the projective space.[29] These algorithms are implemented in systems like GAP and Magma. In GAP, theSylowSubgroup function uses backtrack search over the stabilizer chain derived from Schreier-Sims to construct and conjugate Sylow subgroups in permutation and matrix groups. Magma's equivalent implementation follows the approach of Cannon, Cox, and Holt, incorporating coset enumeration for normalizer computations and supporting both permutation and matrix inputs via internal conversions when needed.[30]