Sylow theorems

In mathematics, specifically 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Peter Ludwig Mejdell Sylow ( 12 December 1832 &ndash 7 September 1918) was a Norwegian Mathematician, who proved foundational Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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.

1 Definition
2 Sylow theorems
3 Example applications
4 Proof of the Sylow theorems
5 Finding a Sylow subgroup
6 References

Definition

Let p be a prime number; then we define a Sylow p-subgroup (sometimes p-Sylow subgroup) of G to be a maximal p-subgroup of G (i. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 e. , a subgroup which is a p-group, and which is not a proper subgroup of any other p-subgroup of G). In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p The set of all Sylow p-subgroups for a given prime p is sometimes written Syl_p(G).

Collections of subgroups which are each maximal in one sense or another are not uncommon in group theory. The surprising result here (see below) is that in the case of Syl_p(G), all members are actually isomorphic to each other and have the largest possible order: if |G| = pⁿs where p does not divide s, then any Sylow p-subgroup P has order |P| = pⁿ. That is, P is a p-group and gcd(|G:P|, p) = 1. These properties can be exploited to futher analyze the structure of G.

The following theorems were first proposed and proven by Norwegian mathematician Ludwig Sylow in 1872, and published in Mathematische Annalen. Peter Ludwig Mejdell Sylow ( 12 December 1832 &ndash 7 September 1918) was a Norwegian Mathematician, who proved foundational Year 1872 ( MDCCCLXXII) was a Leap year starting on Monday (link will display the full calendar of the Gregorian Calendar (or a Leap year The Mathematische Annalen (abbreviated as Math Ann or Math Annal 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. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without

Theorem 1: There exists a Sylow p-subgroup of G, of order pⁿ. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

The following weaker version of theorem 1 was first proved by Cauchy.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G .

Theorem 2: All Sylow p-subgroups of G are conjugate (and therefore isomorphic) to each other, i. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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 n_p be the number of Sylow p-subgroups of G.

n_p divides s.
n_p ≡ 1 mod p.
n_p = |G : N_G(P)|, where P is any Sylow p-subgroup of G and N_G denotes the normalizer. In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which

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.

A very important consequence of Theorem 3 is that the condition n_p = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup. (There are groups which have normal subgroups but no normal Sylow subgroups, such as S₄. )

There is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) which is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn's lemma. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which

Theorem: If K is a Sylow p-subgroup of G, and n_p = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K, and n_p = 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). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. 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 the direct product of groups of order 3 and 5, that is the cyclic group of order 15. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Thus, there is only one group of order 15 (up to isomorphism). In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

A more complex example involves the order of the smallest simple group which isn't cyclic. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning Burnside's p^aq^b theorem states that if the order of a group is the product of two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. In Mathematics, Burnside's theorem in Group theory states that if G is a Finite group of order p^a q^b In Mathematics, a prime power is a Positive integer power of a Prime number. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally 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 = 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 = 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 = 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. In Mathematics, an alternating group is the group of Even permutations of a Finite set. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3. A cyclic Permutation is built from one or more sets of elements in Cyclic order.

Proof of the Sylow theorems

The proofs of the Sylow theorems exploit the notion of group action in various creative ways. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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 p^k has a subgroup of order p^k.

Proof: Let |G| = p^km, and let p^r be the maximal power of p that divides m. Let Ω denote the set of subsets of G of size p^k. Clearly:

$|\Omega | ={p^km \choose p^k}\mathrm{,}$

p^r is precisely the maximal power of p that divides |Ω|, in particular, p^r+1 $\nmid$ ${p^km \choose p^k}$ . Let G act on Ω by left multiplication. It follows by the choice of r that there is an element A ∈ Ω with an orbit θ = GA such that p^r+1 $\nmid$ |θ|. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Now |θ| = |GA| = [G : G_A] where G_A denotes the stabilizer subgroup of the set A, hence p^k | |G_A| so p^k ≤ |G_A|. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. On the other hand, fix an element a ∈ A . The function [g ↦ ga] maps G_A to A injectively; therefore |A| ≥ |G_A|, hence |G_A| = p^k and G_A is a 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|/|G_x| (where G_x denotes the stabilizer), which is a multiple of p by assumption. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. The result follows immediately.

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), i. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g⁻¹Hg = K.

Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH 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 $\nmid$ |Ω₀|, hence in particular |Ω₀| ≠ 0 so there exists some gP ∈ Ω₀. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g⁻¹hgP ⊆ P and therefore g⁻¹Hg ≤ P. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg⁻¹| so that H = gPg⁻¹ for some g ∈ G.

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

Proof: By Theorem 2, n_p = [G : N_G(P)], where P is any such subgroup, and N_G(P) denotes the normalizer of P in G, so this number is a divisor of |G|/q. In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which 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 x ∈ P so that P ≤ N_G(Q). By Theorem 2, P and Q are conjugate in N_G(Q) in particular, and Q is normal in N_G(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 Mathematics, computational group theory is the study of groups by means of computers 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). In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation

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.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents