Sylow theorem

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In mathematics, especially group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the order of G, the existence of corresponding subgroups, and give information about the number of those subgroups.

Definition

Let p be a prime number; then we define a Sylow p-subgroup of G to be a maximal p-subgroup of G (i.e., a subgroup which is a p-group, and which is not a proper subgroup of any other p-subgroup of G). The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).

Collections of subgroups which are each maximal in one sense or another are not uncommon in group theory. The surprising result here is that in the case of Sylp(G), all members are actually isomorphic to each other; and this property can be exploited to determine other properties of G.

Sylow theorems

The following theorems were first proposed and proven by Norwegian mathematician Ludwig Sylow in 1872, and published in Mathematische Annalen. Given a finite group G and a prime p which divides the order of G, we can write the order of G as (pn · s), where n > 0 and p does not divide s.

Theorem 1: There exists a Sylow p-subgroup of G, of order pn.

Theorem 2: All Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), i.e. 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 np be the number of Sylow p-subgroups of G.

  • np divides s.
  • np = 1 mod p.

In particular, the above implies that every Sylow p-subgroup is of the same order, pn; conversely, if a subgroup has order pn, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn. There is also an infinite analog of the Sylow theorems:

Theorem: If K is a Sylow p-subgroup of G, and np = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K, and np = 1 mod p, where Cl(K) denotes the conjugacy class of K.

Example applications

Let G be a group of order 15 = 3 · 5. We have that n3 must divide 5, and n3 = 1 mod 3. The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, n5 divides 3, and n5 = 1 mod 5; thus it also has a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so G must be a cyclic group. Thus, there is only 1 group of order 15 (up to isomorphism), namely Z15.

For a more complex example, we can show that there are no simple groups of order 350. If |G| = 350 = 2 · 52 · 7, then n5 must divide 14 ( = 2 · 7), and n5 = 1 mod 5. Therefore n5 = 1 (since neither 6 nor 11 divides 14), and thus G must have a normal subgroup of order 52, and so cannot be simple.

Proof of the Sylow Theorems

The proofs of the Sylow theorems exploit the notion of group action in various creative ways. The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of H. Wielandt published in 1959. In the following, we use a | b as notation for "a divides b" and a \nmid b for the negation of this statement.

Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk.

Proof: Let |G| = pkm, and let pr be chosen such that no higher power of p divides m. Let Ω denote the set of subsets of G of size pk and note that |Ω| = {p^{k}m \choose p^{k}}{\mathrm  {,}} and furthermore that pr+1 \nmid {p^{k}m \choose p^{k}} by the choice of r. Let G act on Ω by left multiplication. It follows by choice of r that there is an element A ∈ Ω with an orbit θ = AG such that pr+1 \nmid |θ|. Now |θ| = |AG| = [G : GA] where GA denotes the stabilizer subgroup of the set A, hence pk | |GA| so pk ≤ |GA|. Note that the elements gaA for aA are distinct under the action of GA so that |A| ≥ |GA| and therefore |GA| = pk. Then GA is the desired subgroup.

Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω0 denote the set of points of Ω that are fixed under the action of G. Then |Ω| ≡ |Ω0| mod p.

Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω not fixed by G will lie in an orbit of order |G|/|CG(x)| (where CG(x) denotes the centralizer), which is a multiple of p by assumption. The result follows immediately.

Theorem 2: If H is a p-subgroup of a finite group G and P is a Sylow p-subgroup of G then there exists a gG such that HgPg−1. In particular, the Sylow p-subgroups for a fixed prime p are conjugate in G.

Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that |Ω0| ≡ |Ω| = [G : P] mod p. Now p \nmid [G : P] by definition so p \nmid 0|, hence in particular |Ω0| ≠ 0 so there exists some gP ∈ Ω0. It follows that hgP = gP so g−1hgP = P, g−1hgP, and thus hgPg−1hH, so that HgPg−1 for some gG. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg−1| so that H = gPg−1 for some gG.

Theorem 3: Let q denote the order of any Sylow p-subgroup of a finite group G. Then np | |G|/q and np ≡ 1 mod p.

Proof: By Theorem 2, np = [G : NG(P)], where P is any such subgroup, and NG(P) denotes the normalizer of P in G, so this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω0 and observe that then Q = xQx−1 for all xP so that PNG(Q). By Theorem 2, P and Q are conjugate in NG(Q) in particular, and Q is normal in NG(Q), so then P = Q. It follows that Ω0 = {P} so that, by the Lemma, |Ω| ≡ |Ω0| = 1 mod p.

Finding a Sylow subgroup

The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory. In permutation groups, it has been proven by William Kantor that a Sylow p-subgroup can be found in polynomial time of the input (the degree of the group times the number of generators).

References

  • Florian Kammüller and Lawrence C. Paulson. "A Formal Proof of Sylow's Theorem: An Experiment in Abstract Algebra with Isabelle HOL". University of Cambridge, UK. 2000. link
  • H. Wielandt. "Ein Beweis für die Existenz der Sylowgruppen". Archiv der Mathematik, 10:401-402, 1959.fr:Théorèmes de Sylow

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