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Sylow Theorems Sylow Theorems Andrew Clarey Sylow Theorems Definitions/ Looking at the Structure of Arbitrary Groups Theorems Groups, Subgroups Lagrange’s, Normality Class Equation, Cauchy’s Theorem Andrew Clarey First Sylow Theorem Occidental College Theorem Examples Proof Mentor: Professor Nalsey Tinberg Additional Proofs Second Sylow Theorem December 3, 2015 Third Sylow Theorem All material comes from Saracino, Abstract Algebra unless Results Cyclic subgroups otherwise stated. Simple Groups Additional Examples References 1 / 26 Overview Sylow 1 Definitions/ Theorems Theorems Groups, Subgroups Andrew Clarey Lagrange’s, Normality Definitions/ Class Equation, Cauchy’s Theorem Theorems Groups, Subgroups 2 First Sylow Theorem Lagrange’s, Normality Theorem Class Equation, Cauchy’s Theorem Examples First Sylow Theorem Proof Theorem Additional Proofs Examples Proof 3 Second Sylow Theorem Additional Proofs 4 Third Sylow Theorem Second Sylow Theorem 5 Results Third Sylow Cyclic subgroups Theorem Results Simple Groups Cyclic subgroups Additional Examples Simple Groups Additional Examples 6 References References 2 / 26 Definitions/Theorems Sylow Theorems Andrew Clarey A set G is called a group [denoted (G, ∗)] if: Definitions/ i) G has a binary operator ∗. We write a ∗ b as ab. Theorems Groups, Subgroups ii) ∗ is associative Lagrange’s, Normality iii) there is an element e ∈ G such that Class Equation, Cauchy’s Theorem x ∗ e = e ∗ x = x, ∀x ∈ G First Sylow iv) for each x ∈ G, ∃ y ∈ G such that x ∗ y = y ∗ x = e. We Theorem write y = x −1. Theorem Examples Proof A group G is called cyclic if ∃ x ∈ G such that Additional Proofs n G = {x |n ∈ Z} = hxi. Then x is called a generator. Second Sylow Theorem Example cyclic groups are Z, Zn. Third Sylow The order of a group G, denoted |G|, is the number of Theorem Results elements in the group. Cyclic subgroups Simple Groups Additional Examples References 3 / 26 Definitions/Theorems Sylow Theorems A subset H of a group (G, ∗) is called a subgroup of G if Andrew Clarey all h ∈ H form a group under ∗. Definitions/ Theorems Theorem: Let H be a nonempty subset of a group G. Groups, Subgroups Lagrange’s, Then H is a subgroup iff: Normality Class Equation, Cauchy’s Theorem i) ∀a, b ∈ H, ab ∈ H −1 First Sylow ii) ∀a ∈ H, a ∈ H Theorem Theorem We write H ≤ G. Examples If H G, then a Left/Right coset of H in G is a subset Proof 6 Additional Proofs of the form aH/Ha where a ∈ G and Second Sylow Theorem aH/Ha = {ah/ha|h ∈ H}. Third Sylow Two elements x, y ∈ G are conjugate if ∃g ∈ G such that Theorem y = g −1xg. Results −1 Cyclic subgroups If H 6 G, then gHg 6 G is a conjugate subgroup of Simple Groups Additional Examples G, ∀g ∈ G. References 4 / 26 Definitions/Theorems Sylow Theorems Andrew Clarey Lagrange’s Theorem: Let G be a finite group and let Definitions/ H 6 G. Then |H| | |G|, as |G| = |H|[G : H] where [G : H] Theorems is the number of Left/Right cosets. Groups, Subgroups Lagrange’s, Normality Let H 6 G. Then the number of Left/Right Cosets of H Class Equation, Cauchy’s Theorem in G is [G : H], called the index. First Sylow Theorem Let H 6 G. Then we say H is a normal subgroup if Theorem ∀h ∈ H, g ∈ G, ghg −1 ∈ H. We write H G. Examples E Proof Additional Proofs Theorem: Let H 6 G. Then the following are equivalent: Second Sylow Theorem i) H E G Third Sylow −1 Theorem ii) gHg = H, ∀g ∈ G Results iii) gH = Hg, ∀g ∈ G Cyclic subgroups Simple Groups Additional Examples References 5 / 26 Definitions/Theorems Sylow Theorems If H is the only subgroup in G of order |H| then H E G. Andrew Clarey If H E G then G/H is a group called the quotient group Definitions/ whose elements are of the form gH, ∀g ∈ G, and whose Theorems Groups, Subgroups operation is ∗ such that aH ∗ bH = (a ∗ b)H. Lagrange’s, Normality Class Equation, If G, H are groups, then we can define a function φ: Cauchy’s Theorem G → H as a homomorphism if φ(g1g2) = φ(g1)φ(g2). First Sylow Theorem Define a surjection φ from G → G/H where g → gH. Theorem Examples Proof The kernel of φ is given by Ker(φ) = {g ∈ G|φ(g) = eH }, Additional Proofs where eH is the identity in H and it is a normal subgroup. Second Sylow Theorem The Normalizer of H 6 G is the subset Third Sylow N(H) = {g ∈ G|gHg −1 = H}. Theorem Results The Center of a group G is the set of elements Cyclic subgroups Simple Groups Z(G) = {a ∈ G|ag = ga, ∀g ∈ G}. Additional Examples References 6 / 26 Definitions/Theorems Sylow Theorems Andrew Clarey The Centralizer of a g ∈ G is the set of elements Definitions/ Theorems Z(g) = {a ∈ G|ag = ga} Groups, Subgroups Lagrange’s, Normality Theorem: The Class Equation of a group G states: Class Equation, Cauchy’s Theorem |G| = |Z(G)| + [G : Z(g1)] + ··· + [G : Z(gk )], First Sylow g1,..., gk ∈/ Z(G), where each gi is a representative of a Theorem Theorem conjugacy class which contains at least 2 elements. Examples Proof Cauchy’s Theorem: Let G be an abelian group, and let p Additional Proofs be a prime such that p | |G|. Then G contains an element Second Sylow Theorem of order p. That is, ∃x ∈ G so that p is the lowest Third Sylow non-zero number such that x p = e. Theorem Results Cyclic subgroups Simple Groups Additional Examples References 7 / 26 First Sylow Theorem Sylow Theorems Andrew Clarey Definitions/ A subgroup of a group G is called a p-Sylow subgroup if n + n Theorems its order is p , p a prime and n ∈ Z , such that p | |G| Groups, Subgroups n+1 Lagrange’s, and p |G|. Normality - Class Equation, Cauchy’s Theorem First Sylow Theorem: Let G be a finite group, p a First Sylow prime, k ∈ +. Theorem Z Theorem i) If pk | |G|, then G has a subgroup of order pk . In Examples Proof particular, G has a p-Sylow subgroup. Additional Proofs k ii) Let H be any p-Sylow subgroup of G. If K 6 G, |K| = p , Second Sylow −1 Theorem then for some g ∈ G we have K ⊆ gHg . In particular, K Third Sylow is contained in some p-Sylow subgroup of G. Theorem Results Cyclic subgroups Simple Groups Additional Examples References 8 / 26 Examples Sylow Theorems Andrew Clarey Definitions/ 2 4 2 2 Theorems Say |G| = 2 · 3 · 5 · 7 . Then we know there will be at least Groups, Subgroups Lagrange’s, one of each: Normality Class Equation, 2-Sylow subgroup of order 4, Cauchy’s Theorem 3-Sylow subgroup of order 81, First Sylow Theorem 5-Sylow subgroup of order , Theorem Examples 7-Sylow subgroup of order . Proof Additional Proofs We also know there will be subgroups of order 2, 3, 9, 27, 5, Second Sylow and 7. Theorem Third Sylow Theorem Results Cyclic subgroups Simple Groups Additional Examples References 9 / 26 Examples Sylow Theorems 2 Andrew Clarey Let G = A4, a group of order 12 = 2 · 3 Definitions/ Theorems A = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3), (1, 2, 3), (1, 2, 4), Groups, Subgroups 4 Lagrange’s, Normality (1, 3, 4), (1, 3, 2), (1, 4, 3), (1, 4, 2), (2, 3, 4), (2, 4, 3)} Class Equation, Cauchy’s Theorem First Sylow So, a 2-Sylow subgroup of G would be a subgroup of order 4, Theorem Theorem an example is: Examples Proof Additional Proofs H = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)} Second Sylow Theorem Third Sylow In fact this is the only one and therefore is normal, and all Theorem subgroups of order 2 and 4 are contained within it. Results Cyclic subgroups Simple Groups Additional Examples References 10 / 26 Examples Sylow Theorems 1 3 Andrew Clarey Say G = SL2(Z3). Then |G| = 24 = 2 · 3 and −1 ≡ 2mod3. The only 2-Sylow subgroup is: Definitions/ Theorems Groups, Subgroups " # " # " # " # Lagrange’s, 1 0 0 −1 1 1 −1 1 Normality , , , , Class Equation, 0 1 1 0 1 −1 1 1 Cauchy’s Theorem " # " # " # " # First Sylow −1 0 0 1 −1 −1 1 −1 Theorem , , , Theorem 0 −1 −1 0 −1 1 −1 −1 Examples Proof Additional Proofs and there are 4 3-Sylow subgroups: Second Sylow Theorem " # " # " # " # Third Sylow 1 1 1 0 0 1 0 −1 Theorem , , , Results 0 1 1 1 −1 −1 1 −1 Cyclic subgroups Simple Groups Additional Examples References 11 / 26 Proof Sylow Theorems k Andrew Clarey We now prove that if p | |G|, then G has a subgroup of order pk . In particular, G has a p-Sylow subgroup, part i of the First Definitions/ Theorems Sylow Theorem. Groups, Subgroups Lagrange’s, Normality + k Class Equation, Let G be a group, p a prime, k ∈ Z such that p | |G|. We Cauchy’s Theorem will proceed with induction on |G|. If |G| = 2 the result is First Sylow Theorem trivial, and we are done. So, let’s assume the theorem is true Theorem Examples for all groups of order less than |G| and show it is true for |G|. Proof Additional Proofs Second Sylow Case 1: Assume ∃H < G such that p [G : H]. Theorem - |G| = [G : H]|H| so pk must divide |H|. Third Sylow Theorem By the inductive hypothesis, Since |H| < |G|, H has a Results subgroup of order pk , therefore G does as well.
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