In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow^[1] that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.

Corollary — Given a finite group G and a prime number p dividing the order of G, then there exists an element (and thus a cyclic subgroup generated by this element) of order p in G.^[3]

A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, D_2n. For n odd, 2 = 2¹ is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.

By contrast, if n is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/n, half the minimal rotation in the dihedral group.

Another example are the Sylow p-subgroups of GL₂(F_q), where p and q are primes ≥ 3 and p ≡ 1 (mod q) , which are all abelian. The order of GL₂(F_q) is (q² − 1)(q² − q) = (q)(q + 1)(q − 1)². Since q = pⁿm + 1, the order of GL₂(F_q) = p²ⁿ m′. Thus by Theorem 1, the order of the Sylow p-subgroups is p²ⁿ.

Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup.

A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's p^a q^b theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5).

If G is simple, and |G| = 30, then n₃ must divide 10 ( = 2 · 5), and n₃ must equal 1 (mod 3). Therefore, n₃ = 10, since neither 4 nor 7 divides 10, and if n₃ = 1 then, as above, G would have a normal subgroup of order 3, and could not be simple. G then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means G has at least 20 distinct elements of order 3.

As well, n₅ = 6, since n₅ must divide 6 ( = 2 · 3), and n₅ must equal 1 (mod 5). So G also has 24 distinct elements of order 5. But the order of G is only 30, so a simple group of order 30 cannot exist.

Next, suppose |G| = 42 = 2 · 3 · 7. Here n₇ must divide 6 ( = 2 · 3) and n₇ must equal 1 (mod 7), so n₇ = 1. So, as before, G can not be simple.

On the other hand, for |G| = 60 = 2² · 3 · 5, then n₃ = 10 and n₅ = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is A₅, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.

for every prime p. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number n_p of Sylow's p-subgroups in the symmetric group S_p is (p − 2)!. On the other hand, n_p ≡ 1 (mod p). Hence, (p − 2)! ≡ 1 (mod p). So, (p − 1)! ≡ −1 (mod p).

Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow p-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.

The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse,^[4] Scharlau,^[5] Casadio and Zappa,^[6] Gow,^[7] and to some extent Meo.^[8]

The proof will show the existence of some ω ∈ Ω for which G_ω has p^k elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup G_ω, since for any fixed element α ∈ ω ⊆ G, the right coset G_ωα is contained in ω; therefore, |G_ω| = |G_ωα| ≤ |ω| = p^k.

By the orbit-stabilizer theorem we have |G_ω| |Gω| = |G| for each ω ∈ Ω, and therefore using the additive p-adic valuation ν_p, which counts the number of factors p, one has ν_p(|G_ω|) + ν_p(|Gω|) = ν_p(|G|) = k + r. This means that for those ω with |G_ω| = p^k, the ones we are looking for, one has ν_p(|Gω|) = r, while for any other ω one has ν_p(|Gω|) > r (as 0 < |G_ω| < p^k implies ν_p(|G_ω|) < k). Since |Ω| is the sum of |Gω| over all distinct orbits Gω, one can show the existence of ω of the former type by showing that ν_p(|Ω|) = r (if none existed, that valuation would exceed r). This is an instance of Kummer's theorem (since in base p notation the number |G| ends with precisely k + r digits zero, subtracting p^k from it involves a carry in r places), and can also be shown by a simple computation:

and no power of p remains in any of the factors inside the product on the right. Hence ν_p(|Ω|) = ν_p(m) = r, completing the proof.

It may be noted that conversely every subgroup H of order p^k gives rise to sets ω ∈ Ω for which G_ω = H, namely any one of the m distinct cosets Hg.

Lemma — Let H be a finite p-group, let Ω be a finite set acted on by H, and let Ω₀ denote the set of points of Ω that are fixed under the action of H. Then |Ω| ≡ |Ω₀| (mod p).

Any element x ∈ Ω not fixed by H will lie in an orbit of order |H|/|H_x| (where H_x denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately by writing |Ω| as the sum of |Hx| over all distinct orbits Hx and reducing mod p.

Theorem (2) — If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g⁻¹Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), that is, if H and K are Sylow p-subgroups of G, then there exists an element g in G with g⁻¹Hg = K.

Theorem (3) — Let q denote the order of any Sylow p-subgroup P of a finite group G. Let n_p denote the number of Sylow p-subgroups of G. Then (a) n_p = [G : N_G(P)] (where N_G(P) is the normalizer of P), (b) n_p divides |G|/q, and (c) n_p ≡ 1 (mod p).

The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.

One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] is divisible by p, then the normalizer N = N_G(H) of H in G is also such that [N : H] is divisible by p. In other words, a polycyclic generating system of a Sylow p-subgroup can be found by starting from any p-subgroup H (including the identity) and taking elements of p-power order contained in the normalizer of H but not in H itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler,^[10] including the algorithm described in Cannon.^[11] These versions are still used in the GAP computer algebra system.

In permutation groups, it has been proven, in Kantor^[12][13][14] and Kantor and Taylor,^[15] that a Sylow p-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress,^[16] and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.