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Algebra Math Notes • Study Guide Theory

Table of Contents Groups...... 3 Binary Operations ...... 3 Groups ...... 3 Examples of Groups ...... 4 Cyclic Groups ...... 5 and Normal ...... 5 Cosets and Quotient Groups ...... 6 Theorems ...... 7 Product Groups ...... 8 Finite Groups...... 9 and G-sets ...... 9 Combinatorial Equations ...... 10 p-Sylow Subgroups ...... 10 ...... 12 Inner and Outer Automorphisms...... 12 Complete Groups ...... 12 Holomorph ...... 13 Symmetry ...... 14 Symmetry ...... 14 2-D Symmetry Groups ...... 14 3-D Symmetry Groups ...... 15 Groups ...... 16 ...... 16 Transitive Groups ...... 16 Primitive Groups ...... 17 Steiner Systems; Affine and Projective Spaces ...... 18 Mathieu Groups ...... 19 Pólya Enumeration ...... 20 Normal Series ...... 22 Composition Series ...... 22 Commutators and Derived Groups ...... 22 Solvable Groups ...... 23 Hall Subgroups ...... 23 Supersolvable and Nilpotent Groups ...... 24 Frattini ...... 25 Extensions and ...... 26 The Extension Problem ...... 26 Operator Groups...... 27 ...... 27 Cohomology: Background ...... 29 Factor Sets: The Second Cohomology Group ...... 29 Theorems ...... 31 Transfer ...... 31 Derivations: The First Cohomology Group ...... 32 Projective Representations and the ...... 33 Abelian Groups ...... 34 Finite Abelian Groups ...... 34 Infinite Abelian Groups ...... 34 Divisible Groups ...... 35 Pure Subgroups...... 36 Decompositions ...... 36 Subgroups of ℚ ...... 37

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Free Groups ...... 38 Free Groups ...... 38 Todd-Coxeter Algorithm ...... 39 Fundamental Group and Nielsen-Schreier Theorem ...... 40 Free Products and Amalgams ...... 41 HNN Extensions ...... 42 The Word Problem, Decidability of Problems ...... 43 The ...... 44 Linear Groups ...... 45 Classical Groups ...... 45 General ...... 45 SU2 and SO3 ...... 46 One-Parameter Groups ...... 48 Lie ...... 48 Simple Linear Groups ...... 49 ...... 51 The Basics ...... 51 The Bulk of It ...... 52 The Table and Orthogonality Relations ...... 54 Frobenius Reciprocity ...... 55 Applications of Representation Theory ...... 56 References ...... 58

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1 Groups 1-1 Binary Operations

A binary operation on a set S is a rule for combining pairs 푎, 푏 of S to get another element of S (S is closed under the operation), i.e. it defines a map 푆 × 푆 → 푆. Using multiplicative notation, the operation is… 1. associative if 푎푏 푐 = 푎(푏푐). a. If the operation is associative then the product of any n elements (ordered) is well-defined without parentheses. 2. commutative if 푎푏 = 푏푎. 3. Has an identity 1 if 1푎 = 푎, 푎1 = 푎 for all a. a. An element 푎 is invertible if there exists 푎−1 so that 푎푎−1 = 1 = 푎−1푎. b. If 푎 has a left and right inverse, they must be equal. c. An inverse is unique. d. Inverses multiply in opposite : 푎푏 −1 = 푏−1푎−1 Using additive notation, the identity is denoted by 0 and the inverse of 푎 is denoted by – 푎. Exponents 푎푛 become multiples 푛푎. All results will still be true whichever notation is used. Ex. Composition of functions is associative.

1-2 Groups

A group is a set G with an associative binary operation with identity such that every element is invertible. In an , the operation is commutative.

in other words… Groups satisfy 1, 2, 3. Abelian groups also satisfy 4. A semi-group only needs to satisfy 1. 1. For every 푎, 푏, 푐 ∈ 퐺, 푎푏 푐 = 푎 푏푐 . 2. There exists an identity 1 so that 1푎 = 푎1 = 푎 for all 푎. 3. Every element 푎 has an inverse 푎−1 so that 푎푎−1 = 푎−1푎 = 1. 4. For every 푎, 푏 ∈ 퐺, 푎푏 = 푏푎. Note that the existence of right inverses and right identity element imply the existence of the left inverses and left identities (which must be the same).

In a quasi-group, if 푎푏 = 푐, then any two of 푎, 푏, 푐 determine the third uniquely. A loop is a quasi-group with identity.

Quasi-group Semigroup

Loop

Group

Abelian group Cancellation Law: In a group, 푎푏 = 푎푐 or 푏푎 = 푐푎 implies 푏 = 푐.

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A subgroup of a group G is a group H in G, with the same binary operation. In other words, 1. Closure: 푎, 푏 ∈ 퐻 ⇒ 푎푏 ∈ 퐻 2. Identity: 1 ∈ 퐻 3. Inverse: 푎 ∈ 퐻 ⇒ 푎−1 ∈ 퐻. A proper subgroup is not G or {1} (the trivial subgroup). The subgroup 〈푆〉 generated by a set 푆 ⊆ 퐺 is the minimal subgroup in G containing S, i.e. the subgroup of all combinations of the elements in S and their inverses. The intersection of 2 subgroups is a subgroup; the join 퐻 ∨ 퐾 = 〈퐻, 퐾〉 of subgroups 퐻, 퐾 is the minimal subgroup containing both of them. Ex. All subgroups of the additive group of ℤ+ are in the form 푎ℤ = {푎푛|푛 ∈ ℤ} for some 푎.1

The order of a group G, denoted by 퐺 is the number of elements in G.

1-3 Examples of Groups

Groups Description Binary operation ℤ+, ℚ+, ℝ+, ℂ+ Integers, rational numbers, real numbers, and complex numbers2 ℝ×, ℂ× Nonzero real and complex numbers Multiplication 퐓, ℝ/ℤ The : complex numbers of absolute value 1 Multiplication Real numbers modulo 1 Addition 퐶푛 , ℤ푛 , ℤ/(푛) of order n. It can be represented by the Multiplication ℤ/푛ℤ integers modulo 푛, the rotational symmetries of a 푛- Addition gon, or the 푛th roots of unity. Composition Multiplication ℤ(푝∞) Quasicyclic group, p prime. Represented by the rational numbers with a power of p in the denominator, Addition 푘 modulo 1, or the complex numbers 푧 with 푧푝 = 1 for some 푘 ∈ ℕ. Multiplication 퐷푛 of order 2푛. Symmetries of a 푛-gon. Composition 푆푛 on 푛 elements: all permutations of Composition {1,2, … , 푛}. 푆푋 or Perm(푋) denotes the permutations of set 푋. 퐴푛 of order 푛: Permutations with sign Composition +1, or even permutations. 퐻 : {±퐼, ±퐢, ±퐣, ±퐤}, where Multiplication 1 0 푖 0 0 1 0 푖 퐼 = , 퐢 = , 퐣 = , 퐤 = 0 1 0 −푖 −1 0 푖 0 푖2 = 푗2 = 푘2 = −퐼, 푖푗 = −푗푖 = 푘 푗푘 = −푘푗 = 푖, 푘푖 = −푖푘 = 푗 GL푛 (퐹) : 푛 × 푛 invertible matrices over Matrix the F. multiplication

1 This is simple number theory. For a comprehensive development, see Abstract [section number]. 2 + + + ℤ , ℚ , ℝ may also stand for “the positive --- numbers.” The additive groups may be written ℤ+, ℚ+, ℝ+ to avoid confusion (through this is less common). 4

Permutations: A permutation on a set X is a bijective map from a set to itself. A permutation 휋 can be written as a composition of cycles, a cycle written as (푎1푎2 … 푎푘): 휋 푎1 = 푎2, 휋 푎2 = 푎3, … , 휋 푎푘 = 푎1. A permutation has sign +1 (−1) and is even (odd) if any of the equivalent are true: 1. The associated matrix (with 1s at 휋 푖 , 푖 and 0s elsewhere) has determinant +1 (−1). 2. It can be written as the product of an even (odd) number of transpositions. A transposition is a permutation switching 1 pair of elements and leaving everything else fixed. Note that a cycle of 푘 elements can be written as a product of 푛 − 1 transpositions. 3. It has an even number of inversions, pairs 푖 < 푗 with 휋 푖 > 휋(푗). 3 Ex. 푆3 has order 3! = 6 and is generated by 푥 = 123 , 푦 = (12) with the relations 푥 = 1, 푦2 = 1, 푦푥 = 푥2푦.

1-4 Cyclic Groups

The order of an element x is the least positive integer 푛 such that 푥푛 = 1. (If no such n exists, x has infinite order.) The (minimal) exponent of a group G is (the smallest) 푛 such that 푥푛 = 1 for all 푥 ∈ 퐺.

A cyclic group 〈푥〉 is generated by one element x. Its order is the same as the order of the element. Let S be the set of all integers 푘 such that 푥푘 = 1. If the powers of x are not all different, then x has finite order 푛, and 1. 푆 = 푛ℤ, the multiples of n. 2. 푥푟 = 푥푠 iff 푟 − 푠 is a multiple of n. 3. 1, 푥, … , 푥푛−1 are the distinct elements in 〈푥〉.

1-5 Homomorphisms and Normal Subgroups

A from a group 퐺 to a group 퐺′ is a function 휑 such that for all 푎, 푏 ∈ 퐺, −1 −1 휑 푎푏 = 휑 푎 휑(푏). As a consequence, 휑 1퐺 = 1퐺′ and 휑 푎 = 휑 푎 . A homomorphism is an isomorphism if it is bijective: then it has an inverse function that is also a homomorphism. An endomorphism/ is an homomorphism/ isomorphism from a group to itself. Two groups are isomorphic (퐺 ≅ 퐺′) if there is an isomorphism between them. Below, 휑: 퐺 → 퐺′ is a homomorphism.  The of 휑 is the subset of 퐺′ im 휑 = 푥 ∈ 퐺′ 푥 = 휑 푎 for some 푎 in 퐺 .  The of 휑 is the set of elements in G mapped to the identity: ker 휑 = {푎 ∈ 퐺|휑 푎 = 1} Ex. 퐴푛 is the kernel of the sign homomorphism 휍: 푆푛 → ±1 .

The conjugate of an element 푕 ∈ 퐺 by 푔 ∈ 퐺 is 푕푔 = 푔푕푔−1. The conjugate of a subgroup H by 푔 is 퐻푔 = 푔퐻푔−1. Since conjugation is a (injective) homomorphism 푔퐻푔−1 is a subgroup of G as well.  If 푔퐻푔−1 ⊆ 퐻 for all 푔 ∈ 퐺, then 퐻 is normal in G, written 퐻 ⊲ 퐺. (By counting

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elements, equality holds when H is finite, but not necessarily when H is infinite.)  If 휑 퐻 = 퐻 for all automorphisms of G, then H is characteristic in G, written 퐻 char 퐺.  If 휑 퐻 = 퐻 for all endomorphisms of F, then H is a fully invariant subgroup of G. Basic relations  퐻 char 퐾 , 퐾 char 퐺 ⇒ 퐻 char 퐺  퐻 char 퐾 , 퐾 ⊲ 퐺 ⇒ 퐻 ⊲ 퐺

The image is a subgroup of 퐺′ and the kernel is a of 퐺.

1-6 Cosets and Quotient Groups

A left/ right coset of a subgroup 퐻 is a set of the form 푎퐻 = {푎푕|푕 ∈ 퐻} 퐻푎 = {푕푎|푕 ∈ 퐻} respectively, for some 푎 ∈ 퐺. The distinct cosets partition G.  푎퐻 = 푏퐻 iff 푎푏−1 ∈ 퐻.  There are as many left as right cosets. The index of H in G, denoted by [퐺: 퐻], is the number of cosets of 퐻 in G.

Counting Formula: (Lagrange’s Theorem) Let H be a subgroup of G.  퐺 = 퐺: 퐻 퐻 since each coset has |퐻| elements.  퐻 divides |퐺|, and the order of any element divides 퐺 .  퐺: 퐾 = 퐺: 퐻 퐻: 퐾 , for K a subgroup of H. Ex. Any group of prime order is cyclic. Ex. The elements in ℤ푛 relatively prime to 푛 form a group under multiplication. The order of an element divides the order of the group, 휑(푛). For 푛 a prime, the order divides 푛 − 1. The existence of a primitive element shows the group is cyclic.

When H is normal, 푎퐻 = 퐻푎 for all 푎 and the cosets form the 퐺 = 퐺 퐻 with multiplication defined by 푎퐻 푏퐻 = 푎푏퐻. The canonical map 휋: 퐺 → 퐺 sending 푎 ⇝ 푎퐻 is a surjective homomorphism with kernel H.

Let 휑: 퐺 → 퐺′ be a homomorphism. 휑 determines an equivalence relation, with 푎 ≡ 푏 iff 휑 푎 = 휑(푏). This gives a partition of 퐺 into fibers of the map 휑, where each fiber contains all the elements mapped to a single value. The fibers are the cosets of ker 휑 in 퐺. The number of cosets is the number of elements in im 휑.  퐺 = ker 휑 im 휑  | ker 휑 | divides 퐺 .  im 휑 divides both |퐺| and |퐺′ |. Ex. 퐴푛 contains half of the elements of 푆푛 for 푛 > 1.

Index inequalities  퐻: 퐻 ∩ 퐾 ≤ 퐻 ∨ 퐾: 퐾 with equality iff 퐻퐾 = 퐻 ∨ 퐾.  퐻 ∨ 퐾: 퐻 ∩ 퐾 ≤ 퐻 ∨ 퐾: 퐻 퐻 ∨ 퐾: 퐾 with equality iff 퐻퐾 = 퐻 ∨ 퐾.

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퐻 ∨ 퐾 퐾 ≥ 퐻

퐻 ∩ 퐾 For 퐴, 퐵 ⊆ 퐺, 퐺 is a union of disjoint double cosets of the form 퐴푥퐵 = {푎푥푏|푎 ∈ 퐴, 푏 ∈ 퐵}. The order of a double coset is 퐴푥퐵 = 퐴 퐵: 푥퐴푥−1 ∩ 퐵 .

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Correspondence Theorem: Let 휑: 퐺 → 퐺′ be a surjective homomorphism with kernel 퐾.  There is a bijective correspondence between subgroups of 퐺′ and subgroups of 퐺 that contain 퐾. A subgroup of G containing K is associated with its image. G 퐺′ ≅ 퐺/퐾 휑 H 퐻′ ≅ 퐻/퐾

K 1  If 퐻, 퐻′ are corresponding subgroups, 퐻 is normal in 퐺 iff 퐻′ is normal in 퐺′.  퐻 = 퐻′ 퐾

First Isomorphism Theorem: Let 휑: 퐺 → 퐺′ be a homomorphism with kernel 퐾.  The quotient group 퐺 = 퐺/퐾 is isomorphic to the image 퐺′.  (Mapping property) Let 휋: 퐺 → 퐺 be the canonical map. There is a unique homomorphism 휑 : 퐺 → 퐺′ such that 휑 = 휑 ∘ 휋.

퐺 휑 휋 휑 퐺 퐺′

Second Isomorphism Theorem: (Diamond Theorem) Let G be a group. Let H be a subgroup, and N a normal subgroup. Then  The product 퐻푁 = 푕푛 푕 ∈ 퐻, 푛 ∈ 푁 is a subgroup of G.  퐻 ∩ 푁 is a normal subgroup of 퐻.  퐻푁 푁 ≅ 퐻 퐻 ∩ 푁 퐻푁

푁 퐻

퐻 ∩ 푁

Third Isomorphism Theorem: Let G be a group, and N and K be normal subgroups of G with 퐾 ⊆ 푁 ⊆ 퐺. Then  푁 퐾 ⊲ 퐺 퐾  (퐺 퐾) (푁 퐾) ≅ 퐺 푁

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1-8 Product Groups

The product group 퐺 × 퐺′ is the group consisting of the elements 푔, 푔′ , 푔 ∈ 퐺, 푔′ ∈ 퐺′, with coordinatewise multiplication 푎, 푎′ 푏, 푏′ = 푎푏, 푎′ 푏′ . The inclusion homomorphisms 퐺, 퐺′ → 퐺 × 퐺′ are defined by 푖 푎 = 푎, 1 , 푖′ 푏 = (1, 푏) and the homomorphisms 퐺 × 퐺′ → 퐺, 퐺′ are defined by 푝 푎, 푎′ = 푎, 푝′ 푎, 푎′ = 푎′. 푉 = 퐶2 × 퐶2 is called the Klein four group.

Let H, K be subgroups of subgroups of G, and let 푓: 퐻 × 퐾 → 퐺 be the multiplication map 푓 푕, 푘 = 푕푘. 1. 푓 is injective iff 퐻 ∩ 퐾 = {1}. 2. 푓 is a homomorphism iff elements of K commute with elements of H: 푕푘 = 푘푕. 3. 푓 is an isomorphism (퐺 ≅ 퐻 × 퐾) iff 퐻 ∩ 퐾 = {1}, 퐻퐾 = 퐺, and H, K are normal subgroups.

The (a.k.a. unrestricted direct product) of groups 퐺푖, 푖 ∈ 퐼 is the group where each element is an indexed set with 1 member from each group 퐺푖, with coordinatewise multiplication. The direct sum (a.k.a. restricted direct product) is the subgroup of the direct product containing only those elements with a finite number of non-identity elements. The definitions coincide when there are finitely many factors.

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2 Finite Groups 2-1 Permutations and G-sets

Cayley’s Theorem: Every group is isomorphic to a subgroup of a permutation group. Pf. Define 휑: 퐺 → Perm 퐺 (set of permutations of G) by associating 푔 ∈ 퐺 with the permutation of G induced by left multiplication by g.

A group operation of G on a set S is a map 퐺 × 푆 → 푆 such that: 1. 1푠 = 푠 for all 푠 ∈ 푆. 2. 푔푕 푠 = 푔(푕푠) for all 푔, 푕 ∈ 퐺, 푠 ∈ 푆. S is called a G-set. (G may act on the right instead.)

Multiplication by an element of G defines a permutation of the elements of S (they are permutations since 푔−1(푔푠) = 푠). This gives a permutation representation of S, a map 퐺 → Perm(푆).

The orbit of 푠 ∈ 푆 is the set of elements in S that s can be sent to by elements in G: ′ ′ 푂푠 = 푠 푠 = 푔푠 for some 푔 ∈ 퐺 The orbits partition the set S. The operation is transitive if the orbit is the whole group (then any element can get sent to any other element).

The stabilizer of is the subgroup of elements of G that leave s fixed: 퐺푠 = {푔 ∈ 퐺|푔푠 = 푠} ′ ′ Note 퐺 = 퐺푠 푂푠 since for each 푠 ∈ 푂푠, there are 퐺푠 elements 푔 such that 푠 = 푔푠.

Ex. 1: G acts transitively on the set of left cosets of a subgroup H (even when H is not normal) by defining 푎 푏퐻 = 푎푏 퐻. The stabilizer of the coset H is the subgroup H.

Ex. 2: G acts on itself by conjugation: 푔 ∗ 푥 = 푔푥푔−1. The stabilizer of an element x under conjugation is the centralizer: −1 푍퐺 푥 = 푔 ∈ 퐺 푔푥푔 = 푥 = {푔 ∈ 퐺|푔푥 = 푥푔} The centralizer contains and the Z. The orbit of x under conjugation is the conjugacy class of x: 퐶 푥 = 푔푥푔−1 푔 ∈ 퐺 The conjugacy classes partition the group G.

Ex. 3: G acts on the set of subgroups of G by conjugation: 푔 ∗ 퐻 = 푔퐻푔−1. The stabilizer of a subgroup is the normalizer: −1 푁퐺 퐻 = 푔 ∈ 퐺 푔퐻푔 = 퐻 The orbit of H is the set of subgroups conjugate to H. The elements of G leaving each element of H fixed is the centralizer: −1 퐶퐺 퐻 = 푔 ∈ 퐺 푔푕푔 = 푕 for all 푕 ∈ 퐻

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2-2 Combinatorial Equations

Variations of the Same Concept Set of g so that Set of g*h Equation g*h=h (g*H=H) (g*H) possible Left multiplication of G on H Cosets Counting formula subgroup H (Lagrange’s Theorem) 퐺 = 퐻 [퐺: 퐻] Homomorphism 휑 operating (휑 푔푕 = 휑(푕)) (Set of 휑(푔푕)) 퐺 = ker 휑 im 휑 on 푔푕 with 푔, 푕 ∈ 퐺. Kernel Image Group operation (G) on Stabilizer Orbit 퐺 = 퐺푕 푂푕 elements 푕 Orbit equation 퐺 = |푂| orbit 푂 Conjugation of elements Centralizer Conjugacy 퐺 = 푍 푕 퐶 푕 푕 ∈ 퐺 class Class equation 퐺 = |퐶| conjugacy class 퐶 Conjugation of subgroup Normalizer Conjugates of 퐺 = 푁 퐻 퐺: 푁 퐻 퐻 ⊆ 퐺 H

2-3 p-Sylow Subgroups

For p prime, a (finite) p-group is a group whose order is a power of p.  The center of a nontrivial p-group is not trivial. (In the class equation, each term is a power of p. More than one term must be 1; these correspond to elements whose stabilizer is the whole group, i.e. in the center.)  (Fixed Point Theorem) Let G operate on S. If the order of S is not divisible by p, there is a fixed point for the operation of G on S. For G a , if p is prime, 퐺 = 푝푒푚, and H is a subgroup of order 푝푒 , then H is a p- Sylow subgroup (p-SSG) of G.

Sylow Theorems: 1. G contains a p-SSG for all primes p. 푒 푛 a. Let G act on the set S of subsets of order 푝 by conjugation. There are 푝푒 푛 such subsets. Writing the orbit equation for S, since 푝 ∤ 푝푒 , one orbit O has order not divisible by p. The stabilizer of a set U in O has order dividing 푈 = 푝푒 (since U is partitioned into cosets of Stab(푈)) so must be a p-SSG. 2. Any two p-SSG are conjugate in G, and any subgroup of G which is a p-group is contained in some p-SSG. a. Given: p-subgroup K, p-SSG H. Let K operate by left multiplication on the set of cosets of H. There is a fixed point 푔퐻 for this operation. The stabilizer is 푔퐻푔−1, which contains K. 3. If 푛푝 is the number of p-SSGs then 푛푝 ≡ 1 (mod 푝) and divides the order of G. a. Let H operate on the set S of p-SSGs by conjugation. Each orbit has size a power of p, and is divisible by p unless it is a fixed point. If 퐻′ ≠ 퐻 is a fixed

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point, they are both in N(H'), and conjugate in N(H') by (2), contradicting 퐻′ ⊲ 푁(퐻′ ). The are useful in determining all groups of a certain order. (3) can be used to show a group of a certain order is not simple: if there is only 1 p-SSG then it must be normal.

Let P be a p-SSG in G, and let H be a subgroup so that 푁퐺 푃 ⊆ 퐻 ⊆ 퐺. Then 퐻 is its own normalizer.

Frattini Argument: Let K be a normal subgroup of a finite group G. If P is a p-SSG, then 퐺 = 퐾푁퐺 푃 .

If each p-SSG in G is normal in G, then G is the direct product of its p-groups.

Exercises 2 1. Show that every group of order 푝 is abelian, and hence is 퐶푝 or 퐶푝2 . 2. Let 푛 ≥ 5. Given that the only proper normal subgroup of 푆푛 is 퐴푛 , show that the only proper subgroup of index less than n in 푆푛 is 퐴푛 . 3. Let G be a group generated by k elements. There are at most 푛!푘 normal subgroups with index at most n. 4. Let G be a finite group and let p be the smallest prime which divides |G|. If H is a subgroup of G such that [G:H]=p, prove that H is normal in G. 5. Suppose that a finite group has exactly n elements of order p, where p is a prime. Prove that either n=0 or p divides n+1. a. Remark: The following theorem is due to Frobenius: Suppose 퐺 = 푔 and let C be a class of h conjugate elements. The number of solutions to 푥푛 = 푐, 푐 ∈ 퐶 is a multiple of gcd⁡(푕푛, 푔). 6. Let G be a group of order 4n+2. Prove that G is not a .

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3 Automorphisms 3-1 Inner and Outer Automorphisms

The Aut(G) is the group of automorphisms of G under composition. An automorphism is inner if it is conjugation by some element of G; else it is outer. The group of inner automorphisms is Inn(G); Aut(G)/Inn(G) is the .

Basic Theorems: N/C Lemma: If 퐻 ⊆ 퐺 then 퐶퐺(퐻) ⊲ 푁퐺(퐻) and 푁퐺(퐻)/퐶퐺 퐻 can be embedded in Aut(H). Inn(퐺) ⊲ Aut(퐺) and 퐺/푍 퐺 ≅ Inn(퐺). Pf. 퐶퐺(퐻) is the kernel of the group of the restrictions of inner automorphisms (conjugation by an element in 푁퐺(퐻)) to H.

푛 Ex. If G is an of order 푝 then Aut 퐺 ≅ 퐺퐿푛 (픽푝). Aut ℤ ≅ 퐶2. Aut(퐺) = 1 iff 퐺 ≤ 2. Aut 퐶푛 ≅ 푈(ℤ푛 ) where U(R) is the group of invertible elements (units) of the ring R. The group has 휑 푛 elements. 푛−2  Aut ℤ2 = 1 , Aut ℤ4 ≅ ℤ2, Aut ℤ2푚 ≅ ℤ2 × ℤ2푚 −2 . (5 has order 2 )  For an odd prime p, Aut ℤ푝푚 ≅ ℤ 푝−1 푝푚 −1 by primitive roots. 훼1 훼푘 푘  If 푛 = 푝1 ⋯ 푝푘 , then Aut ℤ푛 ≅ 푖=1 Aut(ℤ 훼푖 ). 푝푖

3-2 Complete Groups

A group is complete if it is centerless and every automorphism is inner. Then 퐺 ≅ Aut(퐺).

푆푛 is complete iff 푛 ≠ 2,6. Pf. An automorphism 휑 preserves transpositions iff 휑 is inner. Let 푇푘 be the conjugacy class of k disjoint transpositions. 휑 sends 푇1 to some 푇푘. When 푛 ≠ 2,6, then by counting, 푇1 ≠ 푇푘 , 푘 ≠ 1, so it sends 푇1 to itself. For n=6: 푆5 operates by conjugation on its 6 5-SSGs (5-cycles). The image of the permutation representation (which is an isomorphism, because the kernel is a normal subgroup and hence {1}) is a transitive subgroup K of 120 elements in 푆6. Now let 푆6 operate on the cosets of K. The permutation representation is a bijection (and can be viewed as an automorphism). It can’t preserve the cycle structure of (1 2). Cor. Every outer automorphism of 푆6 switches 푇1, 푇3 so Aut(푆6)/Inn 푆6 ≅ 퐶2 and Aut 푆6 = 1440. Cor. Every finite group imbeds in a complete group. (Actually true for all groups)

Describing outer automorphisms of 푆6: A syntheme is a product of 3 disjoint transpositions; a pentad is a family of 5 synthemes, no two having a common transposition. Two synthemes have a common transposition iff they commute. 푆6 has 6 pentads. An outer automorphism sends each transposition into a syntheme. There are 6 choices of pentads; then 5! choices for how to send {(1 2),…,(1 6)} to the synthemes in a pentad; this accounts for all 720 of them.

If G is a nonabelian simple group, then Aut(G) is complete.

12

Pf. Let 훾푔 be conjugation by g in G and Γ훼 be conjugation by 훼 in Aut(G). Since G is simple and nonabelian, 퐺 ≅ Inn 퐺 = 퐼. A=Aut(G) is centerless- only {1} commutes with every . For any 휍 ∈ 퐴, the subgroup generated by commutators 퐼, 휍 퐼 ∈ 퐼 ∩ 휍 퐼 and must be I (since I is simple). Then 휍 퐼 = 퐼 so 휍 훾푔 = 훾훼 푔 for some −1 −1 automorphism 훼. Then 휏 = 휍Γ훼 is the identity map (check that 휏 훽 훽 ∈ 퐶퐴 퐼 = {1}) so 휍 = Γ훼 .

3-3 Holomorph

Suppose G is mapped to 퐿(퐺) ⊆ 푆(퐺) under the left multiplication permutation representation and 푅(퐺) under the right multiplication permutation. The holomorph Hol 퐺 is defined by Hol 퐺 = 〈퐿(퐺), Aut 퐺 〉 Equivalently, Hol(G) is the normalizer of the subgroup 퐿(퐺) ⊆ 푆(퐺). All automorphisms of G are induced by inner automorphisms of Hol(G).

Properties: 1. 퐿(퐾) ⊲ Hol 퐾 , 퐿(퐾) Aut 퐾 = Hol 퐾 , 퐿(퐾) ∩ Aut 퐾 = {1} 2. Hol 퐾 /퐿(퐾) ≅ Aut(퐾) The subgroup of Hol 퐾 fixing 1 is Aut 퐾 . 3. 푍Hol 퐾 퐿(퐾) = 푅(퐾) 4. Everything works if 퐾푙 and 퐾푟 are switched; left and right multiplication are switched.

If 퐾 ⊲ 퐺 and K is complete, then K is a direct factor of G: there exists a normal subgroup Q of G so that 퐺 = 퐾 × 푄. Conversely, if K is a direct factor whenever it is (isomorphic to) a normal subgroup of a group, then K is complete. 푙 푙 Pf. Take 푄 = 퐶퐺 퐾 . For the converse, identify K with 퐾 ; Hol 퐾 = 퐾 × 퐵 for some B. If 푟 푙 휑 ∈ 퐴푢푡(퐾) then 휑 = 훾푎 for some 푎. Show 퐾 ∩ 퐾 = 푍 퐾 , 퐾 ≅ 퐵 × 푍(퐾). If 휑 is an automorphism of Z(K) then we get an outer automorphism of K. So 푍 퐾 ≤ 2. If it’s 2, embed 퐵 × 푁 ⊂ 퐵 × ℤ4, contradiction. So 푍 퐾 = {1}.

13

4 Symmetry 4-1 Symmetry

An isometry is a rigid motion in ℝ푛, i.e. a distance-preserving map. Every isometry can be uniquely written as the composition of a translation by some vector 푎 and an orthogonal linear operator 휑: 푓 = 푡푎 휑 The set of isometries forms a group 푀푛 .

Change of coordinates: If 휂 changes coordinates, and the formula for the old and new coordinates are 푚 and 푚′, then 푚′ = 휂−1푚휂

A symmetry of a figure, pattern, or set of points F is an isometry that carries F to itself. The symmetries of F form a group. A symmetry 푓=푡푎 휑 preserves orientation if the det 휑 = 1.

Fixed Point Theorem: Let G be a finite group of isometries. Then there is a point in the plane fixed by every element of G. Pf. The centroid of the orbits of any point is sent to itself.

4-2 2-D Symmetry Groups

Isometries of the plane: Type Action Orientation-preserving? Translation 푡푣: 푝 → 푝 + 푣 Yes Rotation Rotate 휃 about some point Yes 휌휃 = reflection around origin Reflection Reflect across some line 푙. No 푟 = reflection across x-axis Glide Reflection across line 푙, No symmetry then translation by nonzero vector parallel to 푙.

Rules: 1. 푡푣푡푤 = 푡푣+푤 , 휌휃 휌휙 = 휌휃+휙 , 푟푟 = 1 ′ 2. 휌휃 푡푣 = 푡푣′ 휌휃 , 푣 = 휌휃 (푣) ′ 3. 푟푡푣 = 푡푣′ 푟, 푣 = 푟(푣) 4. 푟휌휃 = 휌−휃 푟

A does not contain arbitrarily small nonzero rotations and translations, i.e. there exist 휖 > 0 so 1. If translation by 푎 ≠ 0 is in G, then 푎 > 휖. 2. If rotation by 휃 ≠ 0 around some point is in G, then 휃 > 0. In a discrete subgroup of ℝ2, all nonzero vectors have length 푣 > 휖. There is a vector of minimum length.

Tools for analyzing discrete symmetry groups G: 14

1. The translation group L is the set of vectors 푣 such that 푡푣 ∈ 퐺. (Note that the elements are not linear operators.) Every discrete group in ℝ2 is in the form: a. {0} b. ℤ푎, 푎 ≠ 0, or c. ℤ푎 + ℤ푏, {푎, 푏} linearly independent. (A .) 2. Define the homomorphism 휋: 푀푛 → 푂푛 by dropping the translation part of an

isometry: 휋 푡푎 휑 = 휑. The point group is im(휋 퐺). a. A discrete point group can be… 2휋 i. 퐶 , the cyclic group of order 푛 generated by 휌 , 휃 = . 푛 휃 푛 ii. 퐷푛 , the dihedral group of order 2푛 generated by 휌_휃 and 푟. b. Crystallographic Restriction: The point group of the symmetries of a discrete 2 subgroup in ℝ is 퐶푛 or 퐷푛 for 푛 = 1,2,3,4, or 6. (Proof uses that there is a vector of minimal length.) 3. If 푎 ∈ 퐿, 푔 ∈ 퐺 , then 푔 푎 ∈ 퐿.

Discrete symmetry groups (ℝ2): 1. 퐿 = {0}: 퐶푛 or 퐷푛 . 2. 퐿 = ℤ푎: frieze pattern (7 groups) 3. 퐿 = ℤ푎 + ℤ푏: two-dimensional crystallographic group (17 groups)

4-3 3-D Symmetry Groups

Finite subgroups of the rotation group 푆푂3: Group Name and Description Isomorphic Class equation Order to… 퐶푛 Cyclic group 푛 = 1 + ⋯ + 1 n 퐷푛 Dihedral group 2푛 = 1 + ⋯ + 1 + 푛 2n 푇 Tetrahedral group: Rotational 퐴4 12 = 1 + 3 + 4 + 4 12 symmetries of a tetrahedron 푂 Octahedral group: Rotational 24 symmetries of a cube or octahedron 퐼 Icosahedral group: Rotational 퐴5 60 = 1 + 12 + 12 + 15 60 symmetries of dodecahedron or + 20 icosahedron.

Analyze using orbits of poles, unit vectors that are the axis of rotation for some element in the group. (From linear algebra, all orthogonal operators in ℝ3 are rotations.) 230 three-dimensional crystallographic groups, classified into 7 crystal systems, 32 crystallographic point groups.

15

5 Permutation Groups 5-1 Symmetric Group

Two permutations in 푆푛 are conjugate iff the cycles in their cycle decompositions have the same lengths. 휋휍휋−1 has the same cycle decomposition as 휍, but with each 푖 replaced by 휋(푖).

퐴푛 is simple for 푛 ≥ 5. Pf. The class equation of 퐴5 is 60=1+12+12 (2 classes with 5-cycles) +15 (Disjoint 2- cycles) +20 (3-cycles). A normal subgroup must be a union of conjugacy classes and have order dividing 60. Induct for 푛 > 5 using the simplicity criteria in 5-3.

5-2 Transitive Groups

A G-set X with action 휑 is faithful if 휑: 퐺 → 푆푋 is injective. If G has faithful action, G can be called a permutation group. 푋 is the degree of the G-set.

The pointwise stabilizer is 퐺 푥1,…,푥푛 = 푔 ∈ 퐺 푔푥푖 = 푥푖∀푖 (order matters) and the setwise stabilizer is 퐺푌 = 푔 ∈ 퐺 푔 푌 = 푌 where 푌 = {푥1, … , 푥푘 }.

X is transitive if for every 푥, 푦 ∈ 푋 there exists 푔 ∈ 퐺 so that 푦 = 푔푥. Every G-set can be partitioned into transitive G-sets. X is k-transitive if for every pair 푥1, … , 푥푘 , (푦1, … , 푦푘) of k-tuples having distinct entries in X, there exists 푔 ∈ 퐺 so that 푔푥푖 = 푦푖 for each i. For 푘 ≥ 2, X is k-transitive iff for every 푥 ∈ 푋, 푋 − {푥} is (k-1)-transitive. X is sharply k-transitive if only the identity fixes k distinct elements of X.3 A sharply 1-transitive G-set is regular. G is a regular G-set, for any group G.

If 푋 is transitive, then 퐻 = 푔 ∈ 퐺 푔 fixes 푠 is a subgroup of G, and the elements of X can be put into 1-to-1 correspondence with the left cosets of H so that G acts the same way on X as on the left cosets. The 푔 ∈ 퐺 that fix every element forms the largest normal subgroup of G in H.

−1 If 푡푥 = 푦, 푡 ∈ 퐺 then 퐺푦 = 퐺푡푥 = 푡퐺푥 푡 , and X has the same number of 퐺푥 -orbits as 퐺푦 - orbits. The rank of X is the number of 퐺푥 orbits in X, also equal to the number of 퐺푥 푔퐺푥 double cosets in G. Every doubly transitive G-set has rank 2.

If X is k-transitive of degree n, then for every k distinct elements 푥1, … , 푥푘 , 푘 퐺 = 푛 |퐺 푥1,…,푥푘 | 푘 If X is faithful, |퐺 푥1,…,푥푘 | divides 푛 − 푘 ! If X is a sharply k-transitive, 퐺 = 푛 .

The Frobenius kernel N of G is the subset 푁 = 1 ∪ {푔 ∈ 퐺|푔 has no fixed points} If X is a faithful transitive G-set with each 푔 ∈ 퐺 − {1} having at most one fixed point and 1 ⊂ 푁, then G is a . A group G is a Frobenius group iff it has a Frobenius complement, a proper subgroup such that 퐻 ∩ 푔퐻푔−1 = {1} for all 푔 ∈ 퐻. If there

3 These adjectives also apply to groups. We say a group G is [adjective] if there exists a [adjective] G-set. 16

exists a faithful sharply 2-transitive G-set, then G is a Frobenius group.

Frobenius’s Theorem: The Frobenius kernel of a Frobenius group is a normal nilpotent subgroup. (Part of proof in 12-4)

Example. Suppose X is a faithful sharply 2-transitive G-set of degree n. 1. 푁 = 푛 2. If n is odd, then n is a prime power, N is an elementary abelian normal subgroup, and 퐺 = 푁 ⋊ 퐺푥 .

Simplicity Criteria: Let X be a faithful k-transitive G-set, and suppose 퐺푥 is simple for some 푥 ∈ 푋. 1. If 푘 ≥ 4, then G is simple. 2. If 푘 ≥ 3 and 푋 is a power of 2, then 퐺 ≅ 푆3 or G is simple. 3. If 푘 ≥ 2 and 푋 is not a prime power, then G is simple. Application: Constructing larger simple groups (G) from smaller ones (퐺푥 ). A transitive extension of a permutation group on G is a transitive permutation group 퐺 on 푋 = 푋 ∪ {∞}, where ∞ is an element not in G, so that 퐺 ∞ = 퐺.

5-3 Primitive Groups

A block is a subset B of X such that for every 푔 ∈ 퐺, either 푔퐵 = 퐵 or 푔퐵 ∩ 퐵 = 휙. 휙, 푋, {푥} are trivial blocks. A transitive G-set is primitive if it contains no nontrivial blocks. 퐺 The distinct sets among 푔퐵, 푔 ∈ 퐺 form a partition of G. 푌 = 푔퐵 is a transitive G-set, 퐵 the imprimitive system generated by B.

A transitive G-set is primitive iff for each 푥 ∈ 푋, 퐺푥 is a maximal subgroup. Pf. If 퐺푥 ⊂ 퐻 ⊂ 퐺 then 퐻푥 is a nontrivial block.

(In other words, there are no nontrivial G-invariant equivalence relation, i.e. relations with 푥 ≡ 푦 ⇒ 푔푥 ≡ 푔푦. Every doubly transitive G-set X is primitive.)

Jordan’s Theorem: If G is a primitive subgroup of 푆푛 and H is a nontrivial subgroup fixing 푚 points and primitive in its action on the remaining 푛 − 푚 points, then 퐺 is (푚 + 1)-fold transitive. In particular, if G contains a 3-cycle, then 퐺 = 푆푛 or 퐴푛 ; if G contains a transposition, 퐺 = 푆푛 . If H is not primitive, we can still conclude G is doubly transitive. Pf. (1st part) Take a conjugate subgroup 퐻′ of H (distinct from H) that has the most non- fixed points in common with H; if it doesn’t share all but 1 point, then conjugating H by a certain element in 퐻′ gives a subgroup with more non-fixed elements in common. 퐻 ∪ 퐻′ is doubly transitive. Repeat the construction with 퐻 ∪ 퐻′, adding 1 more element each time until you get to G.

A perfect group satisfies 퐺′ = 퐺. If G is a perfect group and X is a faithful primitive G-set, then there is 푥 ∈ 푋 and an abelian normal subgroup 퐾 ⊲ 퐺푥 whose conjugates generate G, then G is simple. Application: Proof of simplicity of PSL’s.

17

5-4 Steiner Systems; Affine and Projective Spaces

Affine Geometry The affine group Aff(푉) of a is the group of functions 푎: 푉 → 푉 for which there is a 푦 ∈ 푉, 푔 ∈ GL(푉) such that 푎 푥 = 푔푥 + 푦. It is a generalization of the group of isometries. If S is a m-dimensional subspace, 푆 + 푣 is an affine m-subspace.

An affine space is a vector space V that’s “forgotten its origin”- think of it as a list of the affine m-subspaces. Considering V as an affine space, Aut(V) is the group of automorphisms preserving affine m-subspaces (i.e. automorphisms such that S is an affine m-subspace iff θ(S) is). The vector space V is a doubly transitive Aff 푉 -space.

A semilinear transformation is a function 푓: 푉 → 푊 satisfying 1. 푓 푥 + 푦 = 푓 푥 + 푓 푦 2. 푓 휆푥 = 휍 휆 푓(푥) for some 휍 ∈ Aut(퐾). (V and W are over the field K.) The bijective semilinear transformations on V form a group Γ퐿(푉) under composition. If dim 푉 ≥ 2, every affine automorphism is in the form 푓 푥 = 푔 푥 + 푢, where 푔 ∈ Γ퐿(푉). 1. Γ퐿 푉 ≅ 퐺퐿(푉) ⋊ Aut(퐾) 2. Aut 푉 ≅ Tr(푉) ⋊ Γ퐿(푉), where Tr⁡(푉) is the translation group.

Projective Geometry For a vector space V over K of dimension n+1, let 푉# = 푉 − {0}, 푥 = 푐푥 푐 ∈ 푉# . The # projective space of V is 푃푛 퐾 = 푃 푉 = 푥 푥 ∈ 푉 ; it has projective dimension n. In # homogeneous coordinates, 푥0, … , 푥푛 = 푐푥0, … , 푐푥푛 , 푐 ∈ 푉 . If W is a (m+1) dimensional subspace of V, 푊 = 푥 푥 ∈ 푊# ⊂ 푃(푉) is a projective subspace of dimension m.  Every pair of distinct points lies on a unique projective line.  If H is a projective hyperplane (of dimension n-1) and L is a projective line not in H, then 퐻 ∩ 퐿 is a projective point.

A collineation (projective isomorphism) is a bijection 휃: 푃 푉 → 푃(푉′) such that S is a projective m-subspace iff 휃(푆) is a projective m-subspace. The projectivity 푃 푔 : 푃 푉 → 푃(푉′) corresponding to the nonsingular semilinear transformation g is 푔 푥 = [푔 푥 ].  Every projective line has 푞 − 1 points. 푛 푛 푘  푃 픽푞 = 푘=0 푞 2 2 2  There are 푞 + 푞 + 1 points and 푞 + 푞 + 1 lines in 푃(픽푞 ).

Fundamental Theorem of Projective Geometry: If 푃 푉 , 푃(푉′ ) have dimension 푛 ≥ 2 (푉, 푉′ have dimension ≥ 3), then every collineation 휃: 푃 푉 → 푃 푉′ is a projectivity.  For 푛 ≥ 2 the group C of collineations of P(V) with itself is isomorphic to 푃Γ퐿(푉) = Γ퐿 푉 푍 푉 , where Z(V) is the center of GL 푉 (nonzero scalar transformations).  PSL(V) acts double transitively on P(V).

A Steiner system of type 푆 푡, 푘, 푣 , 푡 < 푘 < 푣 is an order pair (푋, 퐵) where X is a set with v elements (a.k.a. points), and B is a family of blocks (subsets, a.k.a. lines), each with k elements, so that every t elements of X lie in a unique block. 2 Ex. An affine plane over 픽푞 , with the lines being the blocks, has type 푆(2, 푞, 푞 ). 2 The projective plane 푃2(픽푞 ) has type 푆(2, 푞, 푞 + 푞 + 1).

18

5-5 Mathieu Groups

For a field K, let 퐾 = 퐾 ∪ {∞}. If 휍 ∈ Aut 퐾 and 푎푑 − 푏푐 ≠ 0, then the function 푎휍 휆 + 푏 , 푐휍 휆 + 푑 ≠ 0 푐휍 휆 + 푑 푓 휆 = ∞, 푐휍 휆 + 푑 = 0

∞, 휆 = ∞, 푐 = 0 푎 푐 , 휆 = ∞, 푐 ≠ 0 is a semilinear fractional transformation, and a linear fractional transformation if 휍 is the identity. Γ퐿퐹 퐾 , 퐿퐹(퐾) are the groups of semilinear, linear fractional transformations, respectively. Note Γ퐿퐹 퐾 ≅ 푃Γ퐿2 퐾 , 퐿퐹 퐾 ≅ PGL2(퐾), by associating f with the matrix 푎 푏 . 푐 푑

For 푞 = 푝2푛, p an odd prime, and 휍 the (unique) automorphism of order 2, define the subgroup of Γ퐿퐹(퐾): 푎휆 + 푏 푎휍 휆 + 푏 푀 푞 = 휆 ⇝ 푎푑 − 푏푐 square ∪ 휆 ⇝ |푎푑 − 푏푐 not square 푐휆 + 푑 푐휍 휆 + 푑

Letting 퐾 = 픽푝2푛 , 퐾 is a faithful sharply 3-transitive M(q)-set.

The Mathieu groups are 5 sporadic simple groups. Below, M10 is not considered a , TE=transitive extension, pm=primitive element. Mathieu Order Transitivity Construction4 Automorphism of Reason for group Steiner system simplicity (degree) (system is unique) 4 2 M10 720 = 2 3 5 Sharply 3- M(9) Not simple transitive Acts on 픽9 ∪ {∞}. M11 7920 Sharply 4- Add 휔 to G-set. S(4,5,11) If 퐻 ⊲ 퐺 nontrivial, 4 2  Transitive, 11 divides. = 2 3 5 ⋅ 11 transitive 〈푀10, 푕〉, where  11-SSG of H, 푕 = 휔∞ 휍, 휍 휆 = 푃 ≠ 푁 (푃) 2 3 퐻 휋 휆 + 휋휆 , 휋 pm of 픽9.  푁퐻 푃 = 푁푀11 (푃)  Frattini argument 푀11 = 퐻 M12 95040 Sharply 5- Add Ω to G-set. S(5,6,12) TE of 푀11 6 3 = 2 3 5 ⋅ 11 transitive 〈푀11, 푘〉 where 푘 = 휔Ω 휍, 휍 휆 = 휆3. M22 443520 3- Acts on 푃2 픽4 ∪ {∞}. Subgroup of index TE of 푃푆퐿3 픽4 7 2 = 2 3 5 ⋅ 7 transitive 〈푃푆퐿3 픽4 , 푕1〉, where 2 in Aut(S(3,6,22)) ⋅ 11 푕1 = ∞ 1,0,0 푓1, 2 푓1 푥, 푦, 푧 = [푥 + 푦푧, 푦2, 푧2]. M23 10200960 4- Add 휔 to G-set. S(4,7,23) TE of 푀22 7 2 = 2 3 5 ⋅ 7 transitive 〈푀22, 푕2〉, where ⋅ 11 ⋅ 23 푕2 = 휔∞ 푓2, 푓2 푥, 푦, 푧 = [푥2, 푦2, 훼푧2], 훼 a pm of 픽4.

4 σ, f1, f2, f3 act on 픽9, 푃2 픽4 , 푃2 픽4 , 푃2 픽4 , respectively, and is the identity on everything else. 19

5 M24 244823040 5- Add Ω. S(5,8,24) TE of 푀23 10 3 = 2 3 5 ⋅ 7 transitive 〈푀23, 푕3〉, where ⋅ 11 ⋅ 23 푕3 = Ω휔 푓3, 2 2 2 푓3 푥, 푦, 푧 = [푥 , 푦 , 푧 ] Neither M12 nor M24 have transitive extensions. Sharply transitive groups of degree: 1. All finite groups. 2. Except for a finite number, can be imbedded in Aut 픽푞 . 2푛 3. PGL2 픽푞 , 푀(푝 ) 4. 푆4, 푆5, 퐴6, 푀11 5. 퐴5, 푆6, 퐴7, 푀12 ≥ 6: 푆푘 , 푆푘+1, 퐴푘+2

5-6 Pólya Enumeration

Frobenius-Burnside Lemma: Let 1. 퐺 ⋅ number of orbits = 푔∈퐺 | Fix 푔 |, Fix 푔 = {푠 ∈ 푆|푔푠 = 푠}. a. Both count the number of pairs 푔 ∈ 퐺, 푠 ∈ 푆 such that 푔푠 = 푠. 1 2. number of orbits = | Fix 푔 | 퐺 푔∈퐺

The cycle index of a permutation group G is the average of the cycle index monomials over all permutations g of the group: 1 푍 푎 , 푎 , … = 푎푗 푖 푔 퐺 1 2 퐺 푖 푔∈퐺 푖≥1 where ji(g) is the number of cycles of length i in the disjoint cycle decomposition of g.

Pólya Enumeration Theorem: Let A be a group acting on a set X (the "slots") and and let G operate on YX, the set of all filled slot configurations- each slot in X is filled with an object in Y (the objects, such as beads of different colors, are weighted (1,0,…), (0,1,0,…), …, (0,…,1); the total weight gives the number of each color). Let 퐹퐺 푟, … , 푟푛 be the generating function of the number of orbits of configurations by weight, i.e. 푡1 푡푛 퐹퐺 푟1, … , 푟푛 = 푓퐺 푡1, … , 푡푛 푟1 ⋯ 푟푛 푟1,…,푟푛 where 푓퐺 푡1, … , 푡푛 is the number of orbits with weight (푡1, … , 푡푛 ). Then 2 2 3 3 퐹퐺 푎1, … , 푎푛 = 푍퐺(푎1 + ⋯ + 푎푛 , 푎1 + ⋯ + 푎푛 , 푎1 + ⋯ + 푎푛 , … ) Pf. The only way to color a cycle of length k is to make them all the same color. Use Burnside’s lemma. Generalization: Let G be a group acting on a set X (the "slots") and hence XY, and consider the set YX of all functions from a set X to a weighted set Y (the objects) with weight function ω (the "filled slot configurations"), where the weight of a function f is the sum of the weights of its range. The the generating function of the number of orbits of G on YX by weight (the equivalence classes of configurations induced by X) is given by

5 Application: Golay codes and Leech lattices. 20

2 푛 퐹퐺 푟1, … , 푟푛 = 푍퐺 휔 푦 , 휔 푦 , , … , 휔 푦 푦∈푌 푦∈푌 푦∈푌

Each ω(y) is a monomial in the variables 푟1, 푟2, 푟3 …

휔(푦) = 푓(푟1, 푟2, 푟3, … ) 푦∈푌 where 푓(푟1, 푟2, 푟3, … ) is the generating function of the set Y by weight, so 2 2 3 3 퐹퐺 푎1, … , 푎푛 = 푍퐺 푓 푎1, … , 푎푛 , 푓 푎1 , … , 푎푛 , 푓 푎1 , … , 푎푛 , … Ex. Counting the number of necklaces with n strings of beads on it.

Exercises 1. How many circular necklaces can be made with n beads, if there are c different colors and a. Rotations of necklaces are considered the same. b. Both rotations and reflections of necklaces are considered the same. c. Express the sum in closed form when n is a power of 2.

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6 Normal Series 6-1 Composition Series

A chain of subgroups of G 퐺 = 퐴0 ⊇ 퐴1 ⊇ ⋯ ⊇ 퐴푛 = 1 is a… subnormal (subinvariant) series if each normal (invariant) series if each 퐴푖 is a 퐴푖 is a normal subgroup of 퐴푖−1. normal subgroup of G composition series if each 퐴푖 is a principal (chief) series if each 퐴푖 is maximal maximal normal subgroup in 퐴푖−1. normal subgroup of G contained in 퐴푖−1. In general, a composition series will have fewer terms than a principal series, because there are more requirements.

Two subnormal series are equivalent if the factor groups are the same up to isomorphism and rearrangement. Refining of a subnormal series means to insert more groups between groups in the series.

Schreier Refinement Theorem: Two subnormal series 1 = 퐻0 ⊲ ⋯ ⊲ 퐻푟 = 퐺, 1 = 퐾0 ⊲ ⋯ ⊲ 퐾푠 = 퐺 for the same group G have equivalent refinements. Pf. Use Zessenhaus Lemma: If 퐴 ⊴ 퐵, 퐶 ⊴ 퐷 then 퐴 퐵 ∩ 퐷 퐶 퐷 ∩ 퐵 ≅ 퐴 퐵 ∩ 퐶 퐶 퐷 ∩ 퐴 퐻푖,푗 +1 퐾푖+1,푗 Insert 퐻푖푗 = 퐻푖 퐻푖+1 ∩ 퐾푗 , 퐾푖푗 = 퐾푗 (퐾푗+1 ∩ 퐻푖). Then ≅ . 퐻푖푗 퐾푖푗

Jordan-Hölder Theorem: If G has a composition series S then any subnormal series R can be refined to a composition series, and any two composition series are equivalent. Pf. R and S have equivalent refinements. Removing repeats, these refinements must be equivalent to R. Ex. Taking G to be cyclic of order n, we get that n factors into primes uniquely.

Let H be a normal subgroup of G such that there is a composition series from G to H (in particular, this is true when G is a finite group). Then there is a principal series from G to H where every factor group is the direct product of isomorphic simple groups. Conversely, if such a principal series exists, then there is a composition series.

6-2 Commutators and Derived Groups

A commutator of a group G is in the form 푥, 푦 = 푥푦푥−1푦−1. The commutators generate the derived subgroup 퐺′ of G. The higher order commutators are defined recursively by 푥1, … , 푥푛 = 푥1, … , 푥푛−1 , 푥푛 . For subgroups 퐻1, 퐻2, 퐻1, 퐻2 = 〈(푥1, 푥2)|푥푖 ∈ 퐻푖〉, with higher order commutators defined similarly.

Properties of Commutators: 1. 푦, 푥 = 푥, 푦 −1 2. 푥푦, 푧 = 푥, 푧 푦 푦, 푧 = 푥, 푧 푥, 푧, 푦 (푦, 푧) 3. 푥, 푦푧 = 푥, 푧 푥, 푦 푧 = 푥, 푧 푥, 푧, 푦 푦, 푧

22

4. 푥, 푦−1, 푧 푦, 푧−1, 푥 푧 푧, 푥−1, 푦 푥 = 1 (Jacobi identity) 5. 푥, 푦, 푧 푦, 푧, 푥 푧, 푥, 푦 = 푦, 푥 푧, 푥 푧, 푦 푥 푥, 푦 푥, 푧 푦 푦, 푧 푥 푥, 푧 푧, 푥 푦 6. 푌, 푍, 푋 , 푍, 푋, 푌 ⊆ 퐺 ⇒ 푋, 푌, 푍 ⊆ 퐺

퐺/퐺′ is abelian, and if K is a normal subgroup such that 퐺/퐾 is abelian, then 퐺′ ⊆ 퐾.

(Schur) If 푍(퐺) has finite index in G, then 퐺′ is finite. (See 7-8)

6-3 Solvable Groups

G is solvable if the sequence 퐺 ⊇ 퐺′ ⊇ 퐺′′ ⊇ ⋯ terminates at the identity in a fixed number of steps. Equivalently, G has a solvable series, a subnormal series where all factor groups are abelian (actually, they can be made prime cyclic by refinement). Each of the higher commutator subgroups 퐺 푖 is characteristic in G.

 A finite group is solvable iff the factor groups in a composition series from G to 1 are of prime order.  A finite group is solvable iff the factor groups in a principal series from G to 1 are elementary abelian groups.

Every subgroup and factor group of a is solvable. Conversely, if a finite group G has a normal subgroup H so that H and G/H are both solvable, then G is solvable. In particular, every abelian group is solvable. Pf. If 1 = 퐺푛 ⊆ ⋯ ⊆ 퐺0 = 퐺, then take  1 = 퐻 ∩ 퐺푛 ⊆ ⋯ ⊆ 퐻 ∩ 퐺0 = 퐻  1 = 퐻퐺푛 /퐻 ⊆ ⋯ ⊆ 퐻퐺0/퐻 = 퐺/퐻.  1 = 퐾푛 ⊆ ⋯ ⊆ 퐾0 = 퐺/퐻 gives 퐻 = 퐾푛 퐻 ⊆ ⋯ ⊆ 퐾0퐻 = 퐺. (퐾푖퐻 is the preimage of 퐾푖 under the natural map 퐺 → 퐺/퐻.)

A group that is both solvable and simple must be cyclic of prime order.

Application to Galois Theory: A polynomial equation over a field of characteristic 0 is solvable by radicals iff its Galois group is a solvable group. “Solvable by radicals” means we can get to the roots by repeatedly adjoining (prime) roots of elements in the field to the field; this corresponds to “enlarging” the Galois group by a prime factor each time.

6-4 Hall Subgroups

A H of G is a subgroup whose order and index are relatively prime: gcd 퐻 , 퐺: 퐻 = 1 Let p be a prime. If 퐺 = 푝푛 푎, 푝 ∤ 푎, a p-complement of G is a subgroup with order a.

Extended Sylow Theorems: [Hall] Let G be a solvable group of order 푎푏, where a, b are relatively prime. Then 1. G has a subgroup of order a. 2. Any two subgroups of order a are conjugate, and any subgroup of order dividing a is in a subgroup of order a. 3. The number of subgroups of order a is a product of factors, each of which a. Is congruent to 1 modulo some prime factor of a 23

b. Is a power of a prime and divides one of the chief factors of G. Pf. Induct on |퐺|. Case 1: G has a normal subgroup of order 푎′ 푏′ , 푎′ 푎, 푏′ 푏, 푏′ < 푏. Existence: 퐺/퐻 has subgroup 퐴/퐻 of order 푎/푎′ ; A is the desired subgroup. Conjugacy: Suppose A, A' have order a. Then 퐴퐻/퐻, 퐴′퐻/퐻 ⊂ 퐺/퐻 have order 푎/푎′ and are conjugate. Case 2: Else, take H a minimal normal subgroup. 푏 = 푝푚 ; use the Schur-Zassenhaus lemma.

Converse: If a group contains a p-complement for every prime dividing its order, then G is solvable. Pf. If G is not simple, use the induction hypothesis for G/N (N normal). If G is simple, 푛 푎푖 푛 푎1 푎2 let 퐺 = 푖=1 푝푖 , and let 퐻푖 be a 푝푖-complement. Let 퐷 = 푖=3 퐻푖; then 퐷 = 푝1 푝2 is solvable by Burnside’s theorem. Let N be a minimal normal subgroup of D. 퐺 = 퐻2(퐷 ∩ 퐻1) 퐺 퐺 by comparing orders; 푁 ⊆ 퐻2 ⊂ 퐺, contradiction. (푁 is the minimal normal subgroup of G generated by N.)

6-5 Supersolvable and Nilpotent Groups

The following conditions are stronger than solvability:  A group G is supersolvable if it has a finite normal series 퐺 = 퐴0 ⊇ 퐴1 ⊇ ⋯ ⊇ 퐴푟 = {1}, with each factor group 퐴푖−1/퐴푖 cyclic. Any supersolvable group is finitely generated.  G is nilpotent if it has a , i.e. a finite normal series 퐺 = 퐴0 ⊇ 퐴1 ⊇ ⋯ ⊇ 퐴푟 = {1}, with 퐴푖−1/퐴푖 in the center of 퐺/퐴푖.

1. Define Γ1 퐺 = 퐺, Γ푘 퐺 = 〈 푥1, … , 푥푘 |푥푖 ∈ 퐺〉. Note that Γ푘+1 퐺 = Γ푘 퐺 , 퐺 . The lower central series of G is 퐺 = Γ1 퐺 ⊇ Γ2 퐺 ⊇ ⋯. Each Γ푘 (퐺)/Γ푘+1(퐺) is in the center of 퐺/Γ푘+1(퐺). 2. The upper central series of G is 1 = 푍0 퐺 ⊆ 푍1 퐺 ⊆ ⋯ where 푍푖+1 퐺 /푍푖 퐺 is the center of 퐺/푍푖(퐺). Note each 푍푖 is a . When G is nilpotent, upper and lower central series end at {1} and G, respectively, and have the same length. Then for any central series 퐺 = 퐴0 ⊇ 퐴1 ⊇ ⋯ ⊇ 퐴푟 = {1}, Γ푖 퐺 ⊆ 퐴푖 ⊆ 푍푟+1−푖(퐺) The lower central series is “as fast down as you can go” in a central series, and the upper central series is “as fast up as you can go”.

A group has nil-c if 푥1, … , 푥푐+1 푥푖 ∈ 퐺 = {1}.  If G has nil-c, then every subgroup and quotient group has nil-c.  If H, K are normal subgroups of G, then HK has nil-(c+d).

 Γ푖 퐺 , Γ푗 퐺 ⊆ Γ푖+푗 (퐺) Subgroups and quotient groups of supersolvable groups are supersolvable.

Properties of Nilpotent Groups  Every proper subgroup is strictly contained in its normalizer. (Normalizer condition)  Every maximal subgroup is normal, of prime index, and contains the derived group.  Any abelian group is nilpotent.  Finite p-groups are nilpotent. (Pf. The center of a p-group is not the identity.) 24

 A finite group is nilpotent iff it is the direct product of its Sylow subgroups, which is true iff each p-SSG is normal. (Pf. Take a p-SSG P. 푁퐺(푃) is its own normalizer so by the 1st bullet, is G.)  A finite group is nilpotent iff its maximal subgroups are all normal. (A maximal subgroup is a proper subgroup contained in no other subgroup except G.)

6-5 Frattini Subgroup

The Frattini subgroup Φ(퐺) of G is the intersection of the maximal subgroups of G. It is the set of nongenerators, x such that 〈푇, 푥〉 = 퐺 ⇒ 〈푇〉 = 퐺. Pf. If 푥 ∉ Φ, then 푥 ∉ 푀 for some maximal subgroup M. Then 푀 ⊂ 〈푀, 푥〉 = 퐺. If 푥 ∈ Φ, and 퐺 = 〈푇, 푥〉 ⊃ 〈푇〉, then take the maximal K containing 〈푇〉 but not x. If a maximal subgroup contains 퐾, then it contains x (since 푥 ∈ Φ) and must actually be G, contradiction.

The Frattini subgroup of a finite group is nilpotent. Pf. Take a p-SSG P of G. Since every conjugate of P in Φ is conjugate to P in Φ, 퐺: 푁퐺 푃 = Φ: 푁Φ 푃 ⇒ 퐺: Φ = 푁퐺 푃 : 푁Φ 푃 = 푁퐺 푃 : 푁G 푃 ∩ Φ . But 푁퐺 푃 ∨ Φ: Φ ≥ [푁퐺 푃 ∩ Φ], so 퐺 = 푁퐺 푃 ∨ Φ = 〈푁퐺 푃 , Φ〉. Since Φ consists of nongenerators, 퐺 = 푁퐺(푃).

Burnside Basis Theorem: Let Φ be the Frattini subgroup of the p-group P. 1. The factor group 퐴 = 푃/Φ is an elementary abelian group. 2. If 퐴 = 푝푟, every set of elements which generate P contains a subset of r elements that generate P. (In other words, any minimal generating set has dim 퐺 Φ 퐺 elements.) 3. In the canonical map 푃 → 퐴 they are mapped onto a basis for A. 4. Conversely, any set of r elements which is mapped to a set of generators for A will generate P. 푟 푟−1 푟 푖 푟 5. A has 휃 푝 = 푖=0 푝 − 푝 bases and hence 휃(푝 ) automorphisms. |퐴푢푡 푃 | divides 푝푟 푛−푟 휃(푝푟).

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7 Extensions and Cohomology 7-1 The Extension Problem

The extension of a group K by a group Q is a group G with normal subgroup 퐾1 ≅ 퐾 and factor group 퐺/퐾1 ≅ 푄. In theory terms, there is an 1 → 푁 → 퐺 → 푄 → {1} (each arrow represents a homomorphism; the image of each is the kernel of the next). Why care? Every finite group has a composition series; by knowing the simple groups and the answer to the extension problem we can “recreate” any finite group from the factor groups in the series. Ex. 퐾 × 푄 is an extension of K by Q, but it is by no means the only one because the elements of Q can permute the elements of K under conjugation.

7-2 Direct Products

A group G is decomposable if it is {1} or it can be written as 퐺 = 퐻 × 퐾, with 퐻, 퐾 ≠ {1}. A group has the…  Ascending chain condition (ACC) if there is no infinite strictly increasing chain of normal subgroups 퐾1 ⊂ 퐾2 ⊂ ⋯  Descending chain condition (DCC) if there is no infinite strictly decreasing chain of normal subgroups 퐻1 ⊃ 퐻2 ⊃ ⋯ Ex. ℤ has ACC but not DCC; ℤ 푝∞ has DCC not ACC.

If G satisfies both chain conditions, then G is a direct product of a finite number of indecomposable groups.  If 퐻 ⊲ 퐺, and 퐻, 퐺/퐻 have both chain conditions, so does G.  If 퐻 × 퐾 has both chain conditions, so do 퐻, 퐾.  If G has both chain conditions, a normal endomorphism is an injection iff it is a surjection.

An endomorphism is normal if 휑 푎푥푎−1 = 푎휑 푥 푎−1 for all 푎, 푥 ∈ 퐺. 휑 is nilpotent if there exists 푘 ∈ ℕ so that 휑푘 = 0 (the transformation sending every element to 1).

Fitting’s Lemma: Let G have both chain conditions and 휑 be a normal endomorphism. Then

퐺 = 퐾 × 퐻 for some 휑-invariant subgroups H, K, with 휑 퐾 nilpotent and 휑 퐻 surjective. 푛 푛 Pf. 퐾푛 = ker 휑 , 퐻푛 = im 휑 form an increasing/ decreasing chain stopping at the desired 퐾, 퐻.

Krull-Schmidt Theorem: Let G be a group satisfying both chain conditions. If 퐺 = 퐻1 × ⋯ × 퐻푠 = 퐾1 × ⋯ × 퐾푡 are two decompositions into indecomposable factors, then 푠 = 푡 and there is a reindexing so that 퐻푖 ≅ 퐾푖 for all i. Moreover, given any 푟 (1 ≤ 푟 ≤ 푠), the reindexing may be chosen so that 퐺 = 퐻1 × ⋯ × 퐻푟 × 퐾푟+1 × ⋯ × 퐾푠. Pf. Induct on r. For 푟 = 1, 1. Let 휋푖: 퐺 → 퐻푖, 휆푖: 퐻푖 ↪ 퐺, 휍푗 : 퐺 → 퐾푗 , 휇푗 : 퐾푗 ↪ 퐺 be the projection and inclusion maps. 2. Every partial sum of 휋1 휇푗 휍푗 휆1 is a normal endomorphism of G. 26

3. If 휑, 휓 are nilpotent endomorphisms, and 휑 + 휓 (defined by 휑 + 휓 푥 = 휑 푥 휓(푥)) is an endomorphism, then it is nilpotent by the binomial theorem. This generalizes to more summands. 4. In an indecomposable group, every normal endomorphism if nilpotent or an automorphism. From the contrapositive of (3), and (4), some summand, say 휋1휇1휍1휆1, in (2) is a (normal) automorphism of 퐻1. Then 휍1휆1 is an isomorphism.

7-3 Operator Groups

A set of operators Ω is just a set of endomorphisms of G. (Note, however, that the elements of Ω are allowed to act on different groups, called Ω-groups.) A Ω-map is a homomorphism between Ω-groups invariant under all endomorphisms in Ω: 휑 휔푔 = 휔휑(푔) for all 휔 ∈ Ω, 푔 ∈ 퐺. An admissible subgroup H is invariant under Ω: 휔푕 ∈ 퐻 for all 휔 ∈ Ω, 푕 ∈ 퐻.

Ex. If Ω is the set of inner automorphisms, then the admissible subgroups are the normal subgroups, and a Ω-isomorphism is called a central isomorphism.

The definitions and results about normal subgroups, composition series, and direct products generalize to operator groups:  A group is Ω-simple if it has no proper admissible subgroups.  A Ω-series is a normal series with each term admissible; a Ω-composition series has each factor group Ω-simple.  A group is Ω-decomposable if it is the direct product of nontrivial admissible subgroups.

Jordan-Hölder Theorem: Every two Ω-composition series of a Ω-group are equivalent.

Fitting’s Lemma: Let G have both chain conditions on admissible subgroups and be a

Ω-endomorphism. Then 퐺 = 퐻 × 퐾 for some admissible subgroups H, K, with 휑 퐾 nilpotent

and 휑 퐻 surjective.

Krull-Schmidt Theorem: Let G be a Ω-group satisfying both chain conditions on admissible subgroups. If 퐺 = 퐻1 × ⋯ × 퐻푠 = 퐾1 × ⋯ × 퐾푡 are two decompositions into Ω-indecomposable factors, then 푠 = 푡 and there is a reindexing so that 퐻푖 ≅ 퐾푖 for all i. Moreover, given any 푟 (1 ≤ 푟 ≤ 푠), the reindexing may be chosen so that 퐺 = 퐻1 × ⋯ × 퐻푟 × 퐾푟+1 × ⋯ × 퐾푠.

Ex. The strengthened Jordan-Hölder theorem shows that if a vector space has a finite basis, then the size of a basis is always the same. Fitting’s Lemma shows that if 푇: 푉 → 푉 is a linear transformation for V a finite-dimensional vector space, then 푉 = 푈 ⊕ 푊, for

subspaces U, W with 푇 푈 nilpotent and 푇 푊 nonsingular.

7-4 Semidirect Product

A subgroup 푄 ⊆ 퐺 is a complement of subgroup 퐾 ⊆ 퐺 if 퐾 ∩ 푄 = 1 and 퐺 = 퐾푄. If they exist, complements are unique up to isomorphism. (Warning: Not every normal subgroup 2 has a complement, for example, 〈푎 〉 ⊆ 〈푎〉 = 퐶4. No primary cyclic group is a semidirect 27 product of non-{1} subgroups.)

퐺 is the semidirect product of K by Q 퐺 = 퐾 ⋊ 푄 if 퐾 ⊲ 퐺 and Q is (isomorphic to) a complement of K. G is said to split over K. Each element of Q induces an automorphism of K so this is equivalent to the following formulation: Let θ be a homomorphism of Q into Aut(K); 휃푥 is the image of 푥 ∈ 푄. The semidirect product S of K by Q is the set of pairs 푥, 푢 , 푥 ∈ 퐾, 푢 ∈ 퐻 with the following rule ′ ′ ′ ′ 푥, 푢 푥 , 푢 = (푥푥 , 휃푥′ 푢 푢 ) Connecting the two, 휃푥 is conjugation by 푥 ∈ 푄. We say that G realizes θ; 퐺 = 퐾 ⋊휃 푄. With the above construction, any homomorphism θ can be realized. (Note a group may have distinct factorizations into a semidirect product.)

Ex. 푆푛 = 퐴푛 ⋊ 퐶2 퐷2푛 = 퐶푛 ⋊ 퐶2 Hol 퐾 = 퐾푙 ⋊ Aut(퐾)

Wreath Product

Let D and Q be groups, {퐷휔 |휔 ∈ 훺} a family of groups isomorphic to D, let Q operate on Ω, and let 퐾 = 휔∈훺 퐷휔 . The of D by Q, 퐺 = 퐷 ≀ 푄, is the semidirect product of K by Q, where Q acts on K by 푞 ⋅ 푑휔 = 푑푞휔 , 푞 ∈ 푄. The normal subgroup K is the base of the wreath product G.

Conjugation by an K is D1, D ,… Think of each 푑 ∈ 퐷 as element in Q permutes 2 operating on Λ, the set of “branches” the sets (trees of a tree. Each 푘 ∈ 퐾 operates on the containing certain D1 branches without switching trees. states). Λ

D2

Λ Q operates on indices Q (here Q switches the “trees”)

Ω D3 Example shown: 푆2 ≀ 푆5 is the set of Λ automorphisms of the graph.

The permutation version: If Λ is a D-set, then 훬 × 훺 can be made into a 퐷 ≀ 푄 -set. 푑휆, 휔′ 푖푓 휔′ = 휔 Let 푑∗ 휆, 휔′ = , i.e. 푑∗ operates on one tree, the one indexed by ω. 휔 휆, 휔′ 푖푓 휔′ ≠ 휔 휔 Let 푞∗ 휆, 휔′ = (휆, 푞휔′ ), i.e. q* permutes the trees. ∗ ∗ Then 퐷 ≀ 푄 ≅ 〈푄 , 퐷휔 〉 ⊆ 푆Λ×Ω. If Λ=Q is a Q-set acting on itself, then 푊 = 퐷 ≀푟 푄 is the regular wreath product. 28

If Ω is an infinite set, then the wreath product can be complete or restricted depending on whether the direct product is complete or restricted. (A restricted direct product only include the elements with a finite number of non-1 coordinates.) If Ω and Λ are finite then 푇 ≀ 퐷 ≀ 푄 = (푇 ≀ 퐷) ≀ 푄.

Ex. If p is a prime, then a p-SSG of 푆푝푛 is an iterated wreath product 푊푛 = 퐶푝 ≀푟 ⋯ ≀푟 퐶푝 (n times). Pf. Induct on n. Take 훬 = 푆푝푛 , 푄 = 퐶푝 . Then the resulting wreath product is in 푆훬×훺 ≅ 푆푝푛 +1 and has the right order. Use to calculate a p-SSG of 푆푚 for any m: partition based on base p representation, take p- SSG of each.

7-5 Cohomology: Background

[To be added.]

7-6 Factor Sets: The Second Cohomology Group

From here on, use additive notation in K and G, multiplicative notation in Q, and let 푥푎 denote 푥 operating on 푎. For 퐾 ⊆ 퐺, a right (left) transversal T of K in G is a complete set of right (left) coset representatives- a set containing one element from each right (left) coset. This is less restrictive than a complement.

If 휋: 퐺 → 푄 is surjective, a lifting of 푥 ∈ 푄 is an element 푙 푥 ∈ 퐺 with 휋 푙 푥 = 푥. {푙(푥)|푥 ∈ 푄} is a transversal of ker(π)- each coset is a fiber corresponding to an element in the image.

Let G be an extension of K by Q, and 푙: 푄 → 퐺 a transversal. If K is abelian, there is a homomorphism 휃: 푄 → Aut(퐾) with 휃푥 푎 = 푙 푥 + 푎 − 푙(푥) for every 푎 ∈ 퐾. 휃푥 (conjugation by x) is the same for any transversal l. Pf. Let 휇 = 훾푔 ∈ Aut(퐾) (with the range restricted to K). Since K is abelian, we can define 휇′: 퐺/퐾 → Aut(퐾) by 퐾 + 푔 → 휇(푔). Let 휆: 푄 → 퐺/퐾 be the canonical map defined by ′ 휆 푥 = 퐾 + 푙(푥). 휃푥 = 휇 휆 푥 = 휇 푙 푥 ∈ Aut(퐾) is independent of l. In other words, l(x) can differ by something in K, which commutes with K.

Call an ordered triple (Q,K,θ) data if K is an abelian group (for our purposes), Q is a group, and 휃: 푄 → Aut(퐾) is a homomorphism. G realizes this data if G is an extension of K by Q and for every transversal 푙: 푄 → 퐺, 푥푎 = 휃푥 (푎) = 푙 푥 + 푎 − 푙(푥) for all 푥 ∈ 푄, 푎 ∈ 퐾. (Before, we could state an analogue of this directing by assuming Q is a complement of K; now we introduce transversals to circumvent this!)

Since cosets of K partition G, every element 푔 ∈ 퐺 has a unique expression in the form 푔 = 푎 + 푙 푥 , 푎 ∈ 퐾, 푥 ∈ 푄 Then 푙 푥 + 푙 푦 = 푓 푥, 푦 + 푙(푥푦) 푎 + 푙 푥 + 푏 + 푙 푦 = 푎 + 푥푏 + 푓 푥, 푦 + 푙(푥푦) for some 푓 푥, 푦 ∈ 퐾 because 푙 푥 + 푙 푦 , 푙(푥푦) represent the same coset.

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If 휋: 퐺 → 푄 is a surjective homomorphism with kernel K, and 푙: 푄 → 퐺 is a given transversal with 푙 1 = 0, then the function 푓: 푄 × 푄 → 퐾 defined above is a factor set or cocycle. Basic properties: For all 푥, 푦, 푧 ∈ 푄, 1. 푓 1, 푦 = 0 = 푓 푥, 1 2. Cocycle identity (derived from associativity) 푓 푥, 푦 + 푓 푥푦, 푧 = 푥푓 푦, 푧 + 푓 푥, 푦푧 3. Converse: Given data (Q,K,θ), 푓: 푄 × 푄 → 퐾 is a factor set iff it satisfies the cocycle identity. Construct 퐺푓 by letting 푎, 푥 + 푏, 푦 = (푎 + 푥푏 + 푓 푥, 푦 , 푥푦) 4. If G realizes (푄, 퐾, 휃), and 푙, 푙′ are transversals with 푙 1 = 0 = 푙′ (1) that give rise to factor sets 푓, 푓′, then there is a function 푕: 푄 → 퐾 such that 푕 1 = 0 and 푓′ 푥, 푦 − 푓 푥, 푦 = 푥푕 푦 − 푕 푥푦 + 푕 푥 ∀ 푥, 푦 ∈ 푄

Function Domain and Range f 푄 × 푄 → 퐾 l 푄 → 퐺 h 푄 → 퐾 The factor set measures the deviation of G from being a semidirect product- the obstruction of the transversal 푙 to being a homomorphism. For example, it is identically 0 when G is a semidirect product of K and Q.

A coboundary is a function 푔: 푄 × 푄 → 퐾 for which there exists 푕: 푄 → 퐾 so that 푕 1 = 0 and 푔 푥, 푦 = 푥푕 푦 − 푕 푥푦 + 푕 푥 푍2(푄, 퐾, 휃) is the (abelian) group of all factor sets under addition. 퐵2(푄, 퐾, 휃) is the subgroup of all coboundaries. 퐻2 푄, 퐾, 휃 = 푍2(푄, 퐾, 휃)/퐵2(푄, 퐾, 휃) is the second cohomology group of the data. From (4) above, two extensions 퐺, 퐺′ realizing (푄, 퐾, 휃) are equivalent if they are in the same coset of 퐵2(푄, 퐾, 휃), i.e. if they determine the same element of 퐻2 푄, 퐾, 휃 . Two equivalent extensions are isomorphic, but not necessarily vice versa. Each element in 퐻2 푄, 퐾, 휃 represents an equivalence class of extensions realizing (Q,K,θ). The identity corresponds to the class of the semidirect product.

퐺, 퐺′ are equivalent iff there is an isomorphism γ (dotted line) making the following diagram commute: G {0} K Q {1} G ’ Pf. The isomorphism is 훾 푎 + 푙 푥 = 푎 + 푕 푥 + 푙′ (푥) (푎 ∈ 퐾). We say γ stabilizes the extension; the set of such γ is the stabilizer of the extension.

퐸푥푡(퐾, 푄) is the set of equivalence classes of abelian extensions G of K by Q, where 휃 is the trivial map.

The complete regular wreath product 퐾 ≀푟 푄 contains an isomorphic copy of every 30

extension of K by Q. (K not necessarily abelian)

7-7 Theorems

Schur-Zassenhaus Lemma: A normal Hall subgroup K of a finite group has a complement, so 퐺 = 퐾 ⋊ 퐺/퐾. Any two complements of K in G are conjugate. Pf. 1. An abelian normal Hall subgroup has a complement. (Pf. Sum the cocycle identity 푥푓 푦, 푧 − 푓 푥푦, 푧 + 푓 푥, 푦푧 = 푓(푥, 푦) over 푧 ∈ 푄 = 퐺/퐾. Use Bezout’s Theorem to get 푠 퐾 + 푡 푄 = 1; multiply the sum by 푡 and substitute to get that f is a coboundary with 푕 푥 = 푡 푦∈푄 푓(푥, 푦).) 2. (cont.) Any 2 complements are conjugate. (Pf. We can choose the factor sets to be zero. Sum 0 = 푓1 푥, 푦 − 푓2 푥, 푦 = 푥푕 푦 − 푕 푥푦 + 푕(푥) and use Bezout to get 푕 푥 = 푥푏0 − 푏0; then – 푏0 + 푄1 + 푏0 = 푄2.) 3. For the nonabelian case: If K contains a proper subgroup normal in G, use the induction hypothesis. Else K is a minimal subgroup of G. Using the Frattini argument and the 2nd isomorphism theorem, if P is a p-SSG of K (not {1}), 퐺/퐾 = 푁퐺(푃)/ 푁퐾(푃). If 푁퐺 푃 ⊂ 퐺, then by the hypothesis, 푁퐺 (푃) has a subgroup of order n. Else, 푃 ⊲ 퐺, 퐾 = 푃, 푍 푃 = 푃 and P is abelian; use (1). 4. If either K or G/K is solvable, then any two complements of K in G are conjugate. (Pf. Induction. Suppose 푄1, 푄2 are complements. If K is solvable, and not abelian, then ′ ′ 푄1퐾 /퐾′ is conjugate to 푄2퐾 /퐾′ by induction. If G/K is solvable, let M/K be the minimal normal subgroup of G/K. M=G gives 푄1, 푄2 p-SSGs; else by induction, 푀 ∩ 푄푖 are conjugate. We can assume 퐽 = 푀 ∩ 푄1 = 푀 ∩ 푄2. Then 푄1/퐽, 푄2/퐽 are complements in 퐽(푁퐺 퐽 ∩ 퐾)/퐽 in 푁퐺(퐽)/퐽, and use induction. 5. One of K, G/K is odd and hence solvable by the Feit-Thompson Theorem.

Gaschütz’s Theorem: Let K be a normal abelian p-subgroup of a finite group G, and let P be a p-SSG of G. K has a complement in G iff K has a complement in P. Pf. If Q is a complement of K in G, take 푄 ∩ 푃. If Q is a complement of K in P, choose a transversal U of P in G; then Q+U and -U+Q are transversals. Let f be the factor set of – U+Q. When 푧 = 퐾 + 푞, 푞 ∈ 푄, we have 푓 푥, 푧 = 0 and the cocycle identity becomes 푓 푥, 푦 + 푧 = 푓(푥, 푦). Letting 푡푖 = 푇 = −푈, fixing g and defining 퐾 + 푔 + 푡푖 = 퐾 + 푡휋 푖 + 푞푖, π is a permutation. Sum the cocycle identity over z in {퐾 + 푡|푡 ∈ 푇} and proceed as in (1) above to show f is a coboundary.

7-8 Transfer

If 푄 ⊆ 퐺 is a subgroup of finite index 푛, then the transfer (“Verlagerung”) is the homomorphism 푉: 퐺 → 푄/푄′ (푉퐺→푄) defined by: 푛

푉 푔 = 푥푖푄′ 푖=1 where {푙1, … , 푙푛 } is any left transversal of Q in G and 푔푙푖 = 푙푗 푥푖. (V is independent of the transversal chosen.) If K is a complement, 퐾 ⊆ ker 푉 since all terms in the product would be 1. The transfer is used to find a normal complement K to a normal (p-SSG) subgroup Q. (See Burnside Theorem)

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Useful facts:  If 푄 ⊆ 퐺 is a subgroup of finite index 푛, and 푙푖 , {푕푖} are left transversals, for a fixed 푔 ∈ 퐺 and each 푖, there is a unique 휍(푖) with 푥푖 ∈ 푄, 푔푕푖 = 푙휍 푖 푥푖. σ is a permutation.  For each 푔 ∈ 퐺, there exist 푕푖 ∈ {푙푗 }, 푛푖 ∈ ℕ, 1 ≤ 푖 ≤ 푚, so that −1 푛푖 o 푕푖 푔 푕푖 ∈ 푄, o 푛푖 = 푛 = 퐺: 푄 , −1 푛푖 o 푉 푔 = 푕푖 푔 푕푖 푄′ (Pf. Cycle decomposition of σ)  If Q has finite index 푛 with 푄 ⊆ 푍(퐺), then 푉 푔 = 푔푛 for all 푔.

If a p-SSG if a finite group G has a normal p-complement, G is p-nilpotent.

Burnside Normal Complement Theorem: Let G be a finite group and let Q be an abelian Sylow subgroup contained in the center of its normalizer: 푄 ⊆ 푍 푁퐺 푄 . Then Q has a characteristic complement K. Pf. Take the kernel of the transfer.

Applications: 1. If p is the smallest prime divisor of |퐺|, and G has a cyclic p-SSG, then G is p- nilpotent. (Use that 푁(퐶)/퐶(푄) can be imbedded in Aut(푄).) 2. A nonabelian finite simple group cannot have a cyclic 2-SSG. 3. If every p-SSG of a finite group G is cyclic, then G is solvable. A group of squarefree order is solvable. 4. If G is nonabelian simple, and 푝 is the smallest prime divisor of |퐺|, then 푝3| 퐺 or 12| 퐺 . Using the Feit-Thompson Theorem, every finite simple nonabelian group has order divisible by 8 or 12. 5. (Schur) If 푍(퐺) has finite index in G, then 퐺′ is finite. (퐺′ , 퐺′ ∩ 푍(퐺) are finitely generated; 퐺′ ∩ 푍(퐺) has finite exponent so is finite.)

7-9 Derivations: The First Cohomology Group

Given data (푄, 퐾, 휃), a derivation is a function 푑: 푄 → 퐾 such that 푑 푥푦 = 푥푑 푦 + 푑(푥) A principal derivation is in the form 푑 푥 = 푎 − 푥푎 for some 푎 ∈ 퐾. The kernel of a derivation is the subgroup ker 푑 = {푥 ∈ 푄|푑 푥 = 0}. The set of derivations Der(푄, 퐾, 휃) forms an abelian group under addition; the set of principal derivations PDer 푄, 퐾, 휃 forms a subgroup. The first cohomology group is 퐻1 푄, 퐾, 휃 = Der 푄, 퐾, 휃 / PDer 푄, 퐾, 휃 . Note that the derivation formula comes from the coboundary formula set to 0 (as with semidirect products, when the transversal is a homomorphism). Basic properties: 1. 푑 1 = 0 2. 푑 푥−1 = −푥−1푑 푥 3. 푑 푥 = 푑 푦 ⇔ 푥−1푦 ∈ ker 푑 The stabilizer of an extension G realizing (푄, 퐾, 휃) (i.e. the group of automorphisms fixing each element of K and fixing each coset of K) is isomorphic to Der 푄, 퐾, 휃 . Regarding the

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elements of G as ordered pairs 푎, 푥 , 푎 ∈ 퐾, 푥 ∈ 푄, each stabilizing automorphism is in the form 훾: 푎, 푥 → 푎 + 푑 푥 , 푥 , where d is a derivation.

Let 퐺 = 퐾 ⋊휃 푄, and let 훾: 푎, 푥 → 푎 + 푑 푥 , 푥 be a stabilizing automorphism.  훾 is an inner automorphism iff d is a principal derivation.  퐻1 푄, 퐾, 휃 ⊆ Aut 퐺 Inn 퐺 .  If 퐶, 퐶′ are complements of K in G, and 퐻1 푄, 퐾, 휃 = 0, then 퐶, 퐶′ are conjugate. Application: Another proof of 1&2 in Schur-Zassenhaus. For an abelian normal subgroup of finite index n in a group G, let {푙푖} be a left transversal of K in G. For 푔 ∈ 퐺, let 푒푙푖 = −1 푛 푙휍 푖 휅푖 푒 , 푎푖 푒 = 푙휍 푖 휅푖 푒 푙휍 푖 , 푑 푒 = 푖=1 푎푖 푒 . Gruenberg’s derivation satisfies similar

properties as the transfer; in fact 푑 퐾 is the transfer.

7-10 Projective Representations and the Schur Multiplier

A central extension of K by Q is an extension G of K by Q with 퐾 ⊆ 푍(퐺). The Schur multiplier of Q is the abelian group 푀 푄 = 퐻2 푄, ℂ× = 퐻2(푄, ℂ×, 퐼). The minimal exponent exp 퐺 of a group is the least 푒 ∈ ℕ with 푥푒 = 1 for all 푥 ∈ 퐺. A projective representation of Q is a homomorphism 휏: 푄 → PGL 푛, ℂ = GL 푛, ℂ /푍 푛, ℂ . If K is abelian, the character group of K is 퐾∗ = Hom(퐾, ℂ×).[*] If U is a central extension of K by Q, the transgression of the factor set 푒: 푄 × 푄 → 퐾 is the homomorphism 훿푈: 퐾∗ → 푀(푄) defined by 훿 휑 = [휑 ∘ 푒]. Let 휈: 푈 → 푄 be a surjective homomorphism with kernel K, with U a central extension of K by Q. If 휏: 푄 → PGL 푛, ℂ is a projective representation, then 휏 can be lifted to U if there exists a homomorphism make the diagram commute:

U 휈 Q 휏 휏 휋 G PG U has theL projectiveL lifting property if every projective representation of Q can be lifted to U. A cover of Q is a central extension U of an abelian group K by Q with the projective lifting property and with 퐾 ⊆ 푈′.  If Q is finite, 푀(푄) is finite and exp 푀 푄 | 푄 .  The transgression 훿 is surjective iff U has the projective lifting property.  훿 is injective iff 퐾 ⊆ 푈′.  If Q is finite, 퐵2(푄, ℂ×) has a finite complement 푀 ≅ 푀(푄) in 푍2 푄, ℂ× . Cover Theorem: [Schur] Every finite group Q has a cover U which is a central extension of 푀(푄) by 푄.

If Q is finite, exp 푀 푄 exp 푄 | 푄 . Hence, if every p-SSG of Q is cyclic, then 푀 푄 = 1. Exercises 1. Classify all groups of order pq, p and q prime. In particular, when is there more than one possibility? 2. Find all natural numbers n such that any group of order n is a. cyclic b. abelian

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8 Abelian Groups

We use additive notation and assume the Axiom of Choice.

8-1 Finite Abelian Groups

(Cohn) If A and B are isomorphic abelian groups such that 퐴 = 퐺 × 〈푏〉 and 퐵 = 퐻 × 〈푐〉, where 〈푏〉 ≅ 〈푐〉, then 퐺 ≅ 퐻.

An abelian group corresponds to a ℤ- with integer multiplication defined by 푛푣 = 푣 + 푣 + ⋯ + 푣, so abelian groups and ℤ-modules are equivalent concepts. The 푛 times following is proved using modules (See Notes 5-3):

Structure Theorem for Finitely Generated Abelian Groups: a.k.a Unicity Theorem for Abelian Group Decomposition, Basis Theorem A finitely generated abelian group V is the direct sum of cyclic subgroups and a :

푉 = 퐶푑1 ⊕ ⋯ ⊕ 퐶푑푘 ⊕ 퐿. The orders can be chosen uniquely to satisfy either of the following two conditions:

1. 1 < 푑1|푑2 ⋯ |푑푘 2. All the 푑푖 are prime power orders. (Uniqueness follows from counting orders in p-groups.)

An elementary abelian p-group is a direct sum of cyclic groups of order 푝. If 퐺 is an 푛 elementary abelian p-group of order 푝 , then Aut 퐺 ≅ GL푛(픽푝), since G has the structure of a vector space and homomorphisms correspond to linear transformations.

8-2 Infinite Abelian Groups

A group is p-primary if every element has order a power of p. It generalizes a p-group for abelian groups, to include infinite groups such as ℤ 푝∞ .

The subgroup of an abelian group G is the set of elements with finite order: 푡퐺 = {푥 ∈ 퐺|푛푥 = 0 for some 푛 ∈ ℕ}

A group is  Torsion (periodic) if all elements have finite order (푡퐺 = 퐺).  Torsion-free (aperiodic) if all nonzero elements have infinite order (푡퐺 = 0 ).

Facts:  퐺/푡퐺 is torsion-free; i.e. G is an extension of a torsion group by a torsion-.  Every torsion group is a direct sum of primary groups. 푛  Let 퐺푝 = 푥 ∈ 퐺 푝 푥 = 0 for some 푛 ∈ ℕ0 . Let G and H be torsion groups. 퐺 ≅ 퐻 iff 퐺푝 ≅ 퐻푝 for each prime p. Every finitely generated torsion-free abelian group is free abelian.

Fundamental Theorem: Every finitely generated abelian group G is the direct sum of primary and infinite cyclic groups, and the number of summands depends only on G. 34

Simultaneous Bases: Let H be a subgroup of finite index in a free abelian group of finite rank n. Then there exist bases 푦1, … , 푦푛 ∈ 퐹 and 푕1, … , 푕푛 ∈ 퐻 so that 푕푖 ∈ 〈푦푖〉.

8-3 Divisible Groups

An abelian group G is divisible (complete) if for any 푎 ∈ 퐺 and any natural number 푛, the equation 푛푥 = 푎 has a solution. A group G is reduced if it contains no complete subgroup.

Injective property: Let D be a and A a subgroup of B. If 푓: 퐴 → 퐵 is a homomorphism, then 푓 can be extended to a homomorphism 휑: 퐵 → 퐷. A ↪ B 푖 푓 휑 D Note that the injective property is dual to the projective property in , and a divisible group is like the dual of a free group.  Every factor group of a divisible group is divisible.  The direct sum of divisible groups is divisible.  Any divisible subgroup of a divisible group G is a direct summand of G. Every abelian group can be decomposed into the direct sum of a complete group and a reduced group 퐺 = 퐷 ⊕ 푅. 퐴 is the sum of all divisible subgroups of G, the unique maximal divisible subgroup. (For comparison, G is an extension of by a torsion-free group, but may not be a direct summand.)

푎 The quasicyclic p-group is the p-primary group ℤ 푝∞ = { |푎, 푘 ∈ ℕ} with addition 푝푘 modulo 1. Though it is not cyclic, each of its subgroups is.

Classification of Divisible Abelian Groups: Every divisible group is isomorphic to the direct sum of some number of additive groups of rational numbers and quasicyclic p-groups. + ∞ 퐷 ≅ ℚ + ⋯ + (푝1 + ⋯ ) = 퐻 + 퐹 where F is periodic and H is torsion-free and complete. Pf. Write 퐷 = 푡퐷 ⊕ 푉, V torsion-free. A torsion-free divisible group can be viewed as a vector space over ℚ, (in a torsion-free group, 푛푦 = 푥 has at most one solution for x, so division by natural numbers is well-defined) so is a direct sum of 훿∞ 퐷 = dimℚ 퐷 푡퐷 copies of ℚ. 푡퐷 is a sum of p-primary groups: Letting 퐷 푝 = 푥 ∈ 퐺 푛푥 = 0 , the p-primary ∞ component G of is the sum of 훿푝 퐷 = dimℤ푝 퐷[푝] copies of ℤ 푝 .

Two divisible groups are isomorphic iff they have the same values 훿푝 , 훿∞ (for each p).

Every abelian group can be embedded in a divisible abelian group. G is a direct summand of every abelian group that contains it as a subgroup iff it is divisible.

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8-4 Pure Subgroups

A subgroup H of an abelian group G is a pure (isolated, serving) subgroup if for every 푕 ∈ 퐻 and every natural number n, the equation 푛푥 = 푕 can be solved in H if it can be solved in G. (i.e. ) Ex. A direct summand is pure.

A subgroup B of a torsion group G is basic if 1. B is a direct sum of cyclic groups. 2. B is a pure subgroup of G. 3. G/B is divisible.

Every torsion group G has a basic subgroup, and so is the extension of a direct sum of cyclic groups by a divisible group. Pf. Consider when G is p-primary. G has a pure nonzero cyclic subgroup 〈푦〉- find an element 푥 = 푝푘푦 divisible by 푝푘 but not 푝푘+1 for some k. Take a maximal independent set X with 〈푋〉 pure using Zorn’s lemma. Then 퐺/〈푋〉 is divisible.

H is a pure subgroup of a torsion-free abelian group G iff G/H is torsion-free.

8-5 Direct Sum Decompositions

Let G be a p-primary group.  The elements of the same order form a layer of G. For a p-primary group, the elements of order p form the lowest layer.  An element 푎 has height h if the equation 푝푘 푥 = 푎 has a solution when 푘 ≤ 푕. If it has a solution for all 푘 ≥ 0, 푎 has infinite height.  If every element of the lowest layer of a primary group G has infinite height, then G is divisible.

Kulikov’s Criterion: A primary abelian group G is a direct sum of cyclic groups iff it is the union of an ascending sequence of subgroups 퐴1 ⊆ 퐴2 ⊆ ⋯ such that the elements of each subgroup are of bounded height in G. Pf. Build a well-ordered set of elements of order p until the group it generates contains the lowest layer of G. If it won’t contain the lowest layer, take the one of smallest order not contained; get a contradiction.

Prüfer’s First Theorem: (Prüfer-Baer) Every group with elements of bounded order is a direct sum of cyclic groups. Pf. The only divisible group of bounded order is 0 . So G is basic.

If G is a p-primary group and , then G has 푛 푛+1 푈 푛, 퐺 = dimℤ푝 푝 퐺 ∩ 퐺 푝 푝 퐺 ∩ 퐺 푝 cyclic summands of order 푝푛+1. If G and H are direct sums of p-primary groups, 퐺 ≅ 퐻 iff they have the same Ulm invariants: 푈 푛, 퐺 = 푈(푛, 퐻) for all n.

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 Any 2 basic subgroups of a p-primary group are isomorphic.  If 푡퐺 has bounded order, then 푡퐺 is a direct summand.  A non-divisible torsion group has a p-primary cyclic direct summand, for some p.  An indecomposable group is either torsion or torsion-free.

Prüfer’s Second Theorem: Every countable primary group without elements of infinite height is a direct sum of cyclic groups.  If an abelian group G is a direct sum of cyclic groups, then every subgroup H of G is also a direct sum of cyclic groups.  If an abelian group is a direct sum of cyclic groups, it can be written so uniquely.  Every abelian group is the union of a countable ascending sequence of direct sums of cyclic groups.

A completely decomposable (reducible) group can be written as a of rank 1. It can be written this way uniquely. The unrestricted direct sum of an infinite set of infinite cyclic groups is not completely decomposable.

8-6 Subgroups of ℚ

The rank 휌(퐺) of a torsion-free group G is the number of elements in a maximal independent subset. (Any two maximal independent subsets have the same number of elements.) The rank of an arbitrary abelian group is 휌(퐺 푡퐺).

Every torsion-free group G of rank n can be embedded in a divisible abelian torsion free group ℚ푛 . In particular, every torsion-free group of rank 1 is isomorphic to a subgroup of ℚ.

A characteristic is an infinite sequence of whole numbers or ∞. Two characteristics 푎, 푏 are equivalent (of the same type) if 푎푛 = 푏푛 for all n except a finite number for which they are both finite.

Each abelian torsion-free group of rank 1 corresponds to a type: let 푎 be any nonzero element, write the primes 푝1 < 푝2 < ⋯ in order, and let 푎푛 be the maximum whole number 푎푛 for which 푝푛 푥 = 푎 can be solved. They type determines G up to isomorphism. 1 1 Ex. The subgroup of generated by all the 푎 (when 푎 finite) or 푘 ∈ ℕ (when 푎 infinite) 푖 푖 푝푘 푖 푝푖 푖 has type (푎1, 푎2, … ).

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9 Free Groups 9-1 Free Groups

Take 1: A free group F has a basis X of generators that satisfy no relations other than implied by the group axioms. A word is a finite string of symbols from X or their inverses. The binary operation is juxtaposition. If 푥푥−1 or 푥−1푥 appears in a word, it can be cancelled (into an equivalent word); a word is reduced if no cancellation can be made. This defines a group because… (3 proofs) 1. There is only one reduced form of a word. (Induct on length. If w has 푥푥−1, then one of the 푥’s is cancelled at some stage; cancelling it first has the same effect as cancelling 푥푥−1.) Products of equivalent words are equivalent, and the associative law holds. 2. Van der Waerden trick: For each 푥 ∈ 푋, associate 푥, 푥−1 it with the functions 푥푎 , 푎 = 1, −1 defined by (all exponents below are ±1). 푥푎 푥푎1 ⋯ 푥푎푛 , if 푥푎 ≠ 푥−푎1 푥푎 푥푎1 ⋯ 푥푎푛 = 1 푛 1 1 푛 푎2 푎푛 푎 푎1 푥2 ⋯ 푥푛 , if 푥 = 푥1 Let ℱ be the subgroup of 푆퐹 generated by { 푥 : 푥 ∈ 퐹}. Then 퐹 ≅ ℱ so F is a group.

Let F be the free group generated by 푋, and R a set of words of F. The group generated by X with the relations 푟 = 1, 푟 ∈ 푅 is defined as the quotient group 퐹/푅 where R is the smallest normal subgroup of F containing R. (We impose extra conditions on the elements; and only conditions derived from those need to be satisfied.) The presentation of 퐹/푅 is 〈푥1, … , 푥푛 |푟1, … , 푟푘〉 where 푥푖 are the basis elements in X and 푟푖 are the relations in R.

A free abelian group is the free group with the relations 푥푦푥−1푦−1 = 1 for all letters 푥, 푦 ∈ 푋. Developing it from scratch, it is defined as above but the order of the letters is not important.

Take 2: (Category Theory viewpoint) Mapping (Injective) property of the free group: F is a free group with basis X iff for every group G and every function 푓: 푋 → 퐺, there exists a unique homomorphism 휑: 퐹 → 퐺 extending 푓:

F 휑 inclusion 푓 X G 휑 sends a word in X to the product of the images of the letters in the word. Thus every group G is a quotient of a free group. (Pf. Associate each element in G with an element in X.)

Projective Property: Let 훼: 퐹 → 퐶 be a homomorphism and 훽: 퐵 → 퐶 be a surjective homomorphism. Then there exists a homomorphism 훾: 퐹 → 퐵 such that 훽훾 = 훼:

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F 훾 훼 훽 B C All the above properties hold if “free group” is replaced by “free abelian group” and “group” is replaced by “abelian group.”

Two free groups with bases X, Y are isomorphic iff 푋 = |푌|. The rank of a free group is the number of elements in a basis. If H is a subgroup of the free abelian group F, rank(퐻) ≤ rank(퐹). Pf. By considering X/X', Y/Y' we may assume that X, Y are abelian. 푌 = dim 푌/푝푌 = dim 푋/푝푋 = 푋 . For the last part, let 퐹 = 〈푥푘 |푘 ∈ 퐾〉 (K well-ordered), ′ ′ ′ 퐻푘 = 퐻 ∩ 〈푥푗 푗 < 푘〉, 퐻푘 = 퐻 ∩ 〈푥푗 |푗 ≤ 푘〉. Then 퐻푘 = 퐻푘 or 퐻푘 = 퐻푘 ⊕ 〈푕푘〉. The 푕푘 form a basis for H.

Free semigroups Σ are defined similarly, except there are no inverses and no cancellation. A congruence on a semigroup is an equivalence relation such that 푎 ≡ 푏, 푎′ ≡ 푏′ ⇒ 푎푎′ ≡ 푏푏′. The quotient semigroup Σ/≡ is defined by 푎 푏 = 푎푏 ([푎] is the equivalence class of 푎.) The congruence generated by a subset 퐸 ⊆ Σ × Σ is the intersection of all congruences containing E. The quotient subgroup Σ/≡ has the presentation (푋|푤푖 = 푢푖 for all 푖 ∈ 퐼). If 푤, 푢 ∈ 퐸 then the group has to satisfy 푤 = 푢; it satisfies only those relations implied by these. We cannot write them as 푤푢−1 = 1 because inverses may not exist!

9-2 Todd-Coxeter Algorithm

The Todd-Coxeter Algorithm determines the operation of a finite group on the cosets of a subgroup. Rules for operating on the right cosets of H: 1. The operation of each generator is a permutation. 2. The relations are the identity operator. 3. The generators of H fix H. 4. The operation is transitive.

To compute its group given its presentation 〈푥1, … , 푥푛 |푟1, … , 푟푘〉: 1. Choose a subgroup H. For convenience, use the cyclic subgroup generated by one of the basis elements 푥. (It is possible to use {1} but this often takes too long.) 2. Let 1 represent the coset H. 3. Make tables, one for each relation. Across the top write the letters of the relation in order, and down the columns write the numbers representing cosets. Each letter sends the columns of numbers to the left of it to the column of numbers to the right. For example, here y sends 1 to 2, 2 to 3. letters y z … Numbers representing cosets 1 2 … 2 3 … 4. Start filling in the table with 1. Fill in spaces adjacent to numbers currently in the table. First, see if you can deduce what a letter sends a number to, using the tips below: a. 푥 sends 1 to itself (rule 3), so write x … 39

1 1 b. The entry in the last column of a table matches the first entry by rule (2). c. If you know y sends 푎 → 푏 from somewhere else in the table, then you can fill in y … → y … , y … → y … 푎 ? 푎 푏 ? 푏 푎 푏 The second implication follows from rule (1). 5. If you can’t deduce what a letter sends a number to, then introduce a new number in the next column (for example, we set 푦: 1 → 2 above). For each new number (coset), introduce a new row starting with that number. 6. Collapsing: If at any time 푦: 푎 → 푏 and 푦: 푎 → 푐, then set 푏 = 푐 (erase all c’s and replace them with b’s.) Similarly if 푦: 푎 → 푏 and 푦: 푐 → 푏 then set 푎 = 푐. 7. When all numbers in the table have been accounted for, and no more collapsing can take place, you’re done. (Rule 4 says not to introduce any more cosets.) Find 퐻 (somehow) and use 퐺 = 퐻 퐺: 퐻 to find the order of G. 8. This procedure may not terminate.

9-3 Fundamental Group and Nielsen-Schreier Theorem

Algebraic Topology Background: The Fundamental Group

A complex is a family of nonempty subsets, simplexes (points, edges, triangles, tetrahedrons…), of a set of vertices so that each vertex is a simplex and each nonempty subset of a simplex is a simplex. A simplex with q+1 vertices is a q-simplex; a complex where each simplex has at most n vertices is a n-complex.

A 1-complex is simply a graph; terminology carries over from graph theory. Two paths are homotopic6 (equivalent) if one can be obtained by the other by elementary moves- replacing (u,v)(v,w) by (u,w) where {u,v,w} is a simplex. [푎] denotes the equivalence class of path a.

The fundamental group of a complex K with basepoint v is 휋 퐾, 푣 = {[푎]|푎 is a closed path at 푣} Multiplication is defined by joining the paths together: 푎 푏 = 푎푏 . If K is connected, then the fundamental groups of the different vertices are isomorphic.

A simply connected complex is connected if it is connected and 휋 퐾, 푣 only has the trivial path. A simply connected 1-complex is a tree.

Tietze’s Theorem: If K is a connected complex and T is a maximal tree in K, then 휋 퐾, 푤 ≅ 풯(퐾, 푇) where 풯(퐾, 푇) has the presentation: Generators: Edges 푢, 푣 ∈ 퐾 Relations: (a) 푢, 푣 = 1 for 푢, 푣 ∈ 푇 (b) 푢, 푣 푣, 푥 = (푢, 푥) if {푢, 푣, 푤} is a simplex in K Thus, if K is a connected 1-complex (graph), then it is a free group with rank equal to the number of edges not in T.

6 I.e. One can be continuously deformed into the other (across the 2-simplex, or face). 40

퐾′ is a covering complex of 퐾 if there exists a map 푝: 퐾′ → 퐾 sending simplexes to simplexes, and such that for each simplex s in K, 푝−1(푠) is a disjoint union of simplexes.

Nielsen-Schreier Theorem: Every subgroup of a free group is free. Pf. 1 (Sketch) The fundamental group of a graph K with n cycles meeting at a common point is the free group F with n generators. A subgroup H of F corresponds to a subgroup 휋 of 휋(퐾, 푤), create a covering complex 퐾′ with fundamental group H as follows: 1. The vertices of 퐾′ are equivalence classes of paths in K starting at w, where two paths are equivalent if joining their ends gives a path in 휋: 푎 ≡ 푏 iff they have the same end point and 푎푏−1 ∈ 휋 (푏−1 is the path 푏 in the opposite direction). 2. Two vertices are connected iff there are paths corresponding to the vertices that differ by a path wholly contained in one simplex (i.e. 푏 = 푎푎′ , 푎′ a path in one simplex). 3. The only “meaningful” closed paths in 퐾′ are created by elements that project to 휋, the paths corresponding to elements of H. Since 퐾′ is a graph, its fundamental group is free. Pf. 2 (Sketch) The Cayley graph of a group G with respect to the generators X is the directed graph with vertices the elements of G, with an edge from g to h if there exists 푥, 푕 = 푔푥 for some 푥 ∈ 푋. An automorphism of a graph is a bijective map on the vertices (and edges) sending adjacent vertices to adjacent vertices. Define a metric on the graph: 푑(푥, 푦) is the minimal number of edges in a path from x to y; this extends to subsets. A group G acts freely on a graph (as automorphisms) if the only element of G fixing a vertex or reversing an edge is trivial. Any group acting freely on a locally finite tree (i.e. each vertex has finite degree) is a free group: 1. Take vertex ω, order others in increasing distance. Put 휔 in Γ휔 and at stage k, put vertex 휔푘 in if it is adjacent to 휔푗 , 푗 < 푘 with 휔푗 ∈ Γ휔 , and 휔푘 is not in the G-orbit of an element of Γ휔 . Γ휔 is a subtree with one vertex in each G-orbit. 2. Γ푎 ∩ Γ푏 = 휙 when they are distinct. 3. 푇 = 푡 ∈ 퐺 푑 Γ, 푡Γ = 1 = 푇−1 can be written as a disjoint union 푅 ∪ 푅−1. R is a set of generators for G; if a relation that’s not supposed to be 1 in the free group is 1, then there is a cycle, contradiction. For G free, G acts freely on its Cayley graph, so a subgroup does also, and is free.

If F is a free group of rank n, and H is a subgroup of finite index j, then rank 퐻 = 푗푛 − 푗 + 1. Let H be a subgroup of F. A Schreier transversal of H in F is a right transversal S such 휀1 휀2 휀푛 that whenever the reduced word 푥1 푥2 ⋯ 푥푛 (푥푖 ∈ 푋, 휀푖 = ±1) lies in S, then so does every 휀1 휀2 휀푘 initial segment 푥1 푥2 ⋯ 푥푘 . There exists a Schreier transversal of H in F, and given any −1 Schreier transversal 푆 = {푙(퐻푎)|푎 ∈ 퐹}, then the set of 푕푎,푥 = 푙 퐻푎 푥푙 퐻푎푥 , 푥 ∈ 푋 not equal to 1 form a basis of H.

9-4 Free Products and Amalgams

The of groups ∗푖∈퐼 퐴푖 is the group generated by the groups 퐴푖 when the elements in the different groups are considered to be distinct and which satisfies no relations than those relations in each 퐴푖. In other words, it is the set of all words with letters that are the elements of the 퐴푖, with juxtaposition as multiplication, and where 2 adjacent

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letters can be combined only if they are in the same group 퐴푖. In category theory terms, the free product is the (unique) of the groups 퐴푖, i.e. it satisfies the Universal Mapping Property: The free product is a group P with homomorphisms 푖푗 : 퐴푗 → 푃 (imbeddings) so that for every group G and every family of homomorphisms 푓푗 : 퐴푗 → 퐺 there exists a unique homomorphism 푓 such that 푓푖푗 = 푓푗 for all j. P 푖 푗 푓 푓푗 퐴푗 G

Each 푔 ∈∗푖∈퐼 퐴푖 has a unique factorization called the normal form: 푔 = 푎1 ⋯ 푎푛 where adjacent factors lie in distinct 퐴푖 − 1 .

Each graph is the fundamental group of some 2-complex. Pf. Take a presentation 〈푋|Δ〉, take a graph of 푋 cycles with a common point, each relation is a closed path. Drape a triangulated 2-D cell over each such path. Proof uses Seifert-van Kampen’s Theorem from algebraic topology.

Kurosh Theorem: Let 퐺 =∗푖∈퐼 퐴푖. If 퐻 ⊆ 퐺 is a subgroup, then 퐻 = 퐹 ∗ (∗푗∈퐽 퐻푗 ) for some index set J, where F is a free group and each 퐻푗 is a conjugate of some 퐴푖.  If each 퐴푖 is torsion, and H is torsion-free, then H is a free group.  If H is finite, it is conjugate to a subgroup of some 퐴푖. Every element of finite order is conjugate to an element of finite order in some 퐴푖.

Let 퐴1, 퐴2 be groups having isomorphic subgroups 퐵1, 퐵2, respectively, and let 휃: 퐵1 → 퐵2 be an isomorphism. The amalgam is 퐴1 ∗휃 퐴2 = 퐴1 ∗ 퐴2 푁, where 푁 is the normal −1 subgroup generated by {푏휃 푏 : 푏 ∈ 퐵1}. The amalgam is the (unique) pushout of the diagram 퐵1 퐴푎 푖1 푖2휃 퐴 2 The 푖1, 푖2 are the inclusion maps of 퐵1 → 퐴1, 퐵2 → 퐴2. The pushout is the category theory generalization of union.

Choose a left transversal l of 퐵푖 in 퐴푖, with 푙 1 = 1. Each element 푤푁 ∈ 퐴1 ∗휃 퐴2 has a unique normal form 퐹 푤 푁 = 푤푁, where 퐹 푤 = 푙 푎1 푙 푎2 ⋯ 푙 푎푛 푏, 푏 ∈ 퐵1

and 푙(푎푗 ) lie in transversals of 퐵푖푗 in 퐴푖푗 , and adjacent 푙(푎푗 ) lie in different 퐴푖.

Torsion Theorem: An element in 퐴1 ∗휃 퐴2 has finite order iff it is conjugate to an element of finite order in 퐴1 or 퐴2.

9-5 HNN Extensions

HNN Extensions 1. Let G be a group and let 휑: 퐴 → 퐵 be an isomorphism between subgroups A and B 42

of G. Then the HNN extension 퐺 Ω휑 퐴 with base G and stable letter p is the group with the presentation 〈퐺; 푝|푝−1푎푝 = 휑 푎 for all 푎 ∈ 퐴〉. This group contains G. 2. Let 휑: 퐴푖 → 퐵푖 be , with 휑 푎푖푗 = 푏푖푗 (for some index sets I, J). The group Ω −1 퐸 = 〈퐸; 푝푖 , 푖 ∈ 퐼|푝푖 푎푖푗 푝푖 = 푏푖푗 for all 푖, 푗〉 is an HNN extension with base E and stable letters 푝푖. Visualize the HNN extension of a fundamental group as the fundamental group after topologically “adding a handle” to connect two subcomplexes.

We say that two words are congruent if they have exactly the same spelling before simplifying. If 휔 is a word on 푋 = {푥푖|1 ≤ 푖 ≤ 푛} then 훽 is a subword if there are words 훼, 훾 푒 −푒 so 휔 ≡ 훼훽훾. A pinch in a HNN extension is a word of the form 푤 ≡ 푝푖 푔푝푖 , where either 푒 = −1, 푤 ∈ 퐴푖 ⊆ 퐴 or 푒 = 1, 푤 ∈ 퐵푖 ⊆ 퐵.

Britton’s Lemma: If 휔 is a word with 휔 = 1 in 퐸Ω containing at least one stable letter, then 휔 has a pinch as a subword.

A p-reduced word contains no pinch in the form 푝푒 푔푝−푒 as a subword. Let 퐸Ω be a HNN extension with base E and stable letter p. Then each word on {푋, 푝} has a 푒1 푒푛 unique normal form: a reduced word in the form 휔0푝 휔1 ⋯ 푝 휔푛.

9-6 The Word Problem, Decidability of Group Theory Problems

Does there exist an algorithm that can always decide (in finite time) whether two words in a finitely presented group are the same element? NO. (Novikov-Boone-Britton)

Essentially every algorithm can be run on a Turing machine. The word problem for semigroups Γ (Markov-Post) yields a negative answer because the words can be viewed as configurations on the tape in a Turing machine, and (with special choice of semigroup) the operation of the Turing machine corresponds to “simplifying” the word. By a “diagonalization” argument there exists a recursively enumerable (re) subset S of ℕ (there’s a Turing machine T* that will stop whenever it’s fed an element of S) that is not recursive (ℕ − 푆 is not recursively enumerable, i.e. no algorithm is guaranteed to say if an element is in S). Adapting this gives a non-recursive set 휔 휔 = 휔0 in Γ . An semi-explicit counterexample for groups depending on T* can be given. It is built from multiple HNN extensions to take advantage of Britton’s lemma (pinches). Boone’s lemma says 퐴 Σ = 퐵 Σ ⇔ Σ∗ = 푞 for certain words depending on a word Σ; the connection between these two diagrams can be visualized by supplying relator polygons using the theorem below. Σ∗ is a positive word, so apply the word problem for semigroups to see that there is no decision process for 퐴 Σ 퐵 Σ −1.

Each relation in a presentation of a group can be depicted by a relator polygon, where the edges are labeled with the generators appearing in the relation in order going counterclockwise and arrows point with/ against the flow depending on whether its exponent is 1 or -1.

Fundamental Theorem of Combinatorial Group Theory: Let G have a finite presentation 〈푋|Δ〉 where

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1. Each relation is cyclically reduced (no cyclic permutation has a subword in the form 푥푥−1, 푥−1푥, 푥 ∈ 푋). 2. If 훿 ∈ Δ, then 훿−1 ∈ Δ and every cyclic permutation of δ is in Δ. If 휔 is a cyclically reduced word, then 휔 = 1 iff there is a diagram having a boundary word 휔 and whose regions are relator polygons of relations in Δ. Two elements 휔, 휔′ are conjugate in G iff the annulus with outside/ inside polygon corresponding to 휔 휔′ can be subdivided into relator polygons.

R is recursively presented if it has a presentation 푅 = 〈푢1, … , 푢푚 |휔 = 1, 휔 ∈ 퐸〉 where E is recursively enumerable. Higman Imbedding Theorem: Every recursively presented group can be imbedded in a finitely presented group. Every countable group can be imbedded in a group H having two generators. There exists a universal finitely presented group, a finitely presented group containing a copy of every finitely presented group as a subgroup.

A property M of finitely presented groups is a Markov property if 1. Every group isomorphic to a group with property M has property M. 2. There exists a finitely presented group with property M. 3. There exists a finitely presented group that cannot be imbedded in a finitely presented group with property M. If M is Markov, there does not exist a decision process which will determine for an arbitrary finite presentation, whether the group has property M. This includes: 1. Order 1 2. Finite 3. Finite exponent 4. P-group 5. Abelian 6. Solvable 7. Nilpotent 8. Simple 9. Torsion 10. Torsion-free 11. Free 12. Having a solvable word problem 13. Having a solvable conjugacy problem

9-7 The Burnside Problem

General: If G is a periodic group, and G is finitely generated, then is G necessarily a finite group? NO.

If G is a with exponent n, is G necessarily finite? NO. 푛 Note: It suffices to check whether the free Burnside group 퐵 푚, 푛 = 〈푥1, … , 푥푚 |푥1 = ⋯ = 푛 푥푚 = 1〉 is finite. It is finite for 푛 = 1,2,3,4,6. 푟 + 푟 +푟 Ex. 퐵 푚, 3 = 3 3 2 . Every commutator of weight 4 is the identity.

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10 Linear Groups 10-1 Classical Groups

Name Group Definition General linear group 퐺퐿푛 (퐹) Set of invertible matrices 푆퐿푛 (퐹) {푃 ∈ 퐺퐿푛 (퐹)| det 푃 = 1} Projective group 푃푆퐿푛 (퐹) 푆퐿푛 (퐹)/{±퐼} 푇 푂푛 {푃 ∈ 퐺퐿푛 (ℝ)|푃 푃 = 퐼} Special orthogonal 푆푂푛 {푃 ∈ 푂푛| det 푃 = 1} group ∗ 푈푛 {푃 ∈ 퐺퐿푛 (ℂ)|푃 푃 = 퐼} 푆푈푛 {푃 ∈ 푈푛 | det 푃 = 1} 푆푃2푛 Preserve the skew symmetric form 0 퐼 푆 = ; {푃 ∈ 퐺퐿 (ℝ)|푃푇푆푃 = 푆} −퐼 0 2푛 for 푂푝,푚 Preserves the bilinear form with (p,m)=(3,1) signature (p,m)

A homeomorphism 휑: 푋 → 푌 is a continuous bijective map whose inverse function is also continuous. A manifold of dimension 푑 is a set where every point has a neighborhood homeomorphic to an open set of ℝ푑 .

10-2 General Linear Group

푛 If 퐺 is an elementary abelian 푝-group of order 푝 , then Aut 퐺 ≅ GL푛 (픽푝) , since G has the structure of a vector space and homomorphisms correspond to linear transformations. 푛−1 푛 푛−1 푛 푘 푛 GL푛 픽푞 = 푞 − 푞 = 푞 2 푞 − 1 ⋯ (푞 − 1) 푘=0 푛−1 푛 푛−1 푛 푘 푛 2 SL푛 픽푞 = 푞 − 푞 = 푞 2 푞 − 1 ⋯ (푞 − 1) 푘=1

 The standard Borel subgroup is the subgroup B of all (invertible) upper triangular matrices in the general linear group G.  A matrix is upper unitriangular if it is upper triangular and all entries on the diagonal are 1. They form a group U.  A matrix is unipotent if its characteristic polynomial is 휆 − 1 푛 .7  T is the group of all diagonal matrices.  The Weyl subgroup W is the set of all permutation matrices. 푊 ≅ 푆푛  A transvection (elementary matrix of the first type) 푋푖푗 (푎) is a matrix with 1’s along the diagonal and entry (푖, 푗) equal to 푎. (Multiplication by a transvection adds a multiple of a row/ column to another.) A root subgroup is 푋푖푗 푎 for some fixed 푖, 푗.

Bruhat Decomposition: 퐺 = 퐵푊퐵, and G is the disjoint union of the 푛! double cosets

7 Or 1 − 휆 푛 , depending on how you define characteristic polynomial. 45

퐵푤퐵, 푤 ∈ 푊.

The general linear group is generated by the set of invertible diagonal matrices and all transvections; the special linear group is generated by transvections.

Parabolic subgroups 푛 푛 A flag on 퐹 is an increasing chain of subspaces 0 ⊂ 푊1 ⊂ ⋯ ⊂ 푊푟 = 퐹 . A complete flag has 푛 subspaces- one for each dimension 1 to n. The standard flag has 푊푖 = span(푒1, … , 푒푖) , 1 ≤ 푖 ≤ 푛. A subflag is a flag containing some of the 푊푖. G acts on the (complete) flags by 푔 푊1, … , 푊푟 = (푔푊1, … , 푔푊푟). For the complete flags, the action is transitive and the stabilizer of the standard flag is the Borel subgroup.

퐵 = 푈 ⋊ 푇 (triangular = upper unitriangular ⋊ diagonal) Kolchin’s Theorem: Any unipotent subgroup of G is conjugate with a subgroup of U.

A parabolic subgroup P is the stabilizer of some flag on 퐹푛 . Choose subspaces so 푊푖 = 푊푖−1 ⊕ 푌푖. The unipotent radical of P is the subgroup 푈푃 of matrices that induce the identity transformation on each 푊푖 푊푖−1. A Levi complement 퐿푃 of 푈푃 is the common

stabilizer of the 푌푖; it is isomorphic to GL푦1 (퐹) × ⋯ × GL푦푟 (퐹), where 푦푖 = dim 푊푖 − dim 푊푖−1. 푃 = 푈푃 ⋊ 퐿푃, 푃 = 푁퐺(푈푃)

A staircase group is a parabolic subgroup of G stabilizing some subflag of the standard flag. Ex. The staircase group of 0 ⊂ 푊2 ⊂ 푊3 ⊂ 푊6 consists of matrices of the form ∗ ∗ ∗ ∗ ∗ ∗ 1 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 ∗ ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ , the unipotent radical 0 0 1 , and a Levi complement 0 0 0 ∗ ∗ ∗ 0 0 0 1 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 1 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 1 ∗ ∗ 0 0 0 0 ∗ ∗ 0 0 0 0

0 0 ∗ 0 0 0 . 0 0 0 ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ The only subgroups of G containing B are the staircase groups.

The center Z of GL is the multiples of the identity matrix, and the center of SL is 푍 ∩ 푆퐿.

10-3 SU2 and SO3

SU2 contains exactly the matrices in the form 푎 푏 푃 = , 푎 푎 + 푏 푏 = 1 −푏 푎 or 푥0 + 푥1푖 푥2 + 푥3푖 2 2 2 3 푃 = , 푥0 + 푥1 + 푥2 + 푥3 = 1 −푥1 + 푥2푖 푥0 − 푥1푖 With the correpondence

46

1 0 푖 0 0 1 0 푖 퐼 = , 퐢 = , 퐣 = , 퐤 = 0 1 0 −푖 −1 0 푖 0 this becomes the quaternion algebra. 3 4 The matrices in SU2 correspond to points on the unit 3-sphere 핊 in ℝ . 1. The latitudes of 핊3 are given by 2 2 3 2 푥0 = 푐, 푥1 + 푥2 + 푥3 = 1 − 푐 Each latitude is a 2-dimensional sphere. A latitude corresponds to the conjugacy class of matrices with trace 2c. a. The equator 피 is the latitude with 푥0 = 0; it contains matrices with trace 0. 푇 Each matrix on the equator is associated with the unit vector 푥1, 푥2, 푥3 ∈ ℝ3. 2. The longitudes of 핊3 are 푊 ∩ 핊3, 푊 a 2 − dimensional subspace of ℝ4 containing I They have the parameterization 푙 휃 = cos 휃 퐼 + sin 휃 퐴, 퐴 ∈ 피. The longitudes are conjugate subgroups of SU2. I Diagon al matrice s Equator: Trace 0 k j i matrices

SO

-I 2

Conjugation by an element of SU2 rotates the equator (a 2-D sphere). For an element P in SU2, let γ(P) be the (matrix corresponding to) this rotation (with respect to the basis i, j, k). Then the spin homomorphism 훾: 푆푈2 → 푆푂3 is surjective with kernel {±퐼}. If 푃 = cos 휃 퐼 + sin 휃 퐴, 퐴 ∈ 피, then 훾푃 has spin (A,2θ). γ is a 2-to-1 mapping (antipodal, or opposite points have the same image) so SU2 is a double covering of SO3: SO3 ≅ SU2/Z = SU2/{±I} 3 SO3 ≅ ℙ (projective 3 − space)

Pf. Define a bilinear form for elements of the equator as follows: let 〈U, V〉 = u1v1 + u2v2 + u3v3, where U = u1i + u2j + u3k, V = v1i + v2j + v3k, i.e use the standard dot product with 1 the coordinates. Then 〈U, V〉 = − trace UV . γ P is orthogonal because 〈γ U, γ V〉 = 〈U, V〉. 2 P P The determinant of γP is always 1 by continuity. So the image is in SO3. Just verify γPj = cos 2θ j + sin 2θ k for P = cos θ I + sin θ i; (2) holds by conjugation.

47

10-4 One-Parameter Groups

Recall (see Linear Algebra, 9.1) ∞ 퐴푡 푛 푒퐴푡 = 푛! 푖=0 1. The series converges for any A. 2. If A and B commute, 푒퐴+퐵 = 푒퐴푒퐵. In particular, 푒푡1퐴푒푡2퐴 = 푒 푡1+푡2 퐴. 푑 3. 푒푡퐴 = 퐴푒푡퐴 푑푡

A one-parameter group is a differentiable homomorphism + 휑: ℝ → 퐺퐿푛 퐴푡 These homomorphisms are 휑 푡 = 푒 , for some 퐴 ∈ 퐺퐿푛 (ℂ), and the images of one- parameter groups are exactly those in the form {푒퐴푡 |푡 ∈ ℝ}. 푑휑 푑휑 Pf. Let 퐴 = . Then = 퐴휑. 푑푡 푡=0 푑푡

Finding one-parameter groups in a subgroup G of GLn: 1. Require that 푒퐴푡 ∈ 퐺 for all t. Differentiate to get possibilities for 푑 퐴 = 푒퐴푡 푑푡 푡=0 2. Check to see whether all these matrices A work. Examples: Group G Property Used One-parameter groups in G are 푒퐴푡 with A: 퐴 푂푛 If A is real skew-symmetric, then 푒 is Real skew-symmetric orthogonal. 퐴 푈푛 If A is skew-Hermitian, then 푒 is Skew-Hermitian unitary. trace 퐴 퐴 푆퐿푛 푒 = det 푒 Trace zero matrix

10-5 Lie Algebras

A Lie algebra is a real vector space with a bracket operation 푉 × 푉 → 푉 denoted by [푣, 푤] such that 1. 푐푣1 + 푣2, 푤 = 푐 푣1, 푤 + [푣2, 푤] Bilinearity (one implies 푣, 푐푤1 + 푤2 = 푐 푣, 푤1 + [푣, 푤2] the other) 2. 푣, 푤 = −[푤, 푣] Skew symmetry 3. 푢, 푣, 푤 + 푣, 푤, 푢 + 푤, 푢, 푣 = 0 Jacobi Identity

A vector v is tangent to a subset S of ℝ푘, or a set of matrices, at a point (matrix) x if there is 푑휑 a differentiable path 휑 in S such that = 푣 and 휑 푡0 = 푥 for some 푡0. The Lie 푑푡 푡=푡0 algebra of a subgroup 퐺 ⊂ 퐺퐿푛 (ℂ) is the set of tangent vectors to G at I, with the bracket defined by 퐴, 퐵 = 퐴퐵 − 퐵퐴

A is in the Lie algebra of G iff 푒퐴푡 is a one-parameter subgroup in G. 48

10-6 Simple Linear Groups

Left multiplication (translation) 푚푃 by an element P of a matrix group G is a homeomorphism. Homogeneity is the property that the group “looks the same” everywhere.

A subgroup of 퐺퐿푛 (ℝ) which is a closed subset of 퐺퐿푛 (ℝ) is a manifold. 푛 푛−1 Ex. The orthogonal group 푂 is a manifold of dimension , using the fact that the 푛 2 exponential 푒퐴 maps a small neighborhood U of 0 in ℝ푛×푛 homeomorphically to a neighborhood of I in 퐺퐿푛 (ℝ).

If 퐻 ⊆ 퐺 is a subgroup of the path-connected matrix group G that contains a nonempty open subset U, then 퐻 = 퐺. Pf. 1. The union of cosets of U in H is H. Since the union of an arbitrary number of open sets is open, H is open. 2. A path connected set is not the disjoint union of nonempty open sets: If 푆 = 푈0 ∪ 푈1 and p is a path connecting 푈0 and 푈1, sup 푥 푝 푥 ∈ 푈0 can’t be in either set. 3. G is the union of cosets of H, so 퐺 = 퐻.

푆푂3 ≅ 푆푈2/ ±1 is a simple group. Pf. Let N be a normal subgroup of 푆푂3 1. For 푃0 ∈ 푁, the conjugacy class (latitude) C of 푃0 is in N. −1 2. Take a nontrivial path 푃(푡) from 푃0 to 푃1 in C. Let 푄 푡 = 푃 푡 푃0 . Then Q is in G and goes from I to some other matrix. Q goes through all conjugacy classes sufficiently close to I, so N contains a neighborhood of I and 푁 = 푆푂3.

If 푞 is a power of a prime, 3 1. 푆퐿2 픽푞 = 푞 − 푞 3 2. 푃푆퐿2 픽푞 = 푞 − 푞 if q is a power of 2 1 3. 푃푆퐿 픽 = (푞3 − 푞) else 2 푞 2 Some relationships: 푃푆퐿2 픽2 ≅ 푆3, 푃푆퐿2 픽3 ≅ 퐴4, 푃푆퐿2 픽4 ≅ 푃푆퐿2 픽5 ≅ 퐴5, 푃푆퐿2 픽9 ≅ 8 퐴6, 푃푆퐿2 픽7 ≅ 퐺퐿3(픽2)

If F is a field of order at least 4, 1. The only normal subgroup of 푆퐿2(퐹) is the center 푍 = {±1}. 2. 푃푆퐿2(퐹) is simple. Pf. Let N be a normal subgroup. 1. Choose s so that 푠2 ≠ ±1,0 −1 −1 2. For any 퐴 ∈ 푁, there is a matrix 푃 ∈ 푆퐿2(퐹) so that the commutators 퐴푃퐴 푃 ∈ 푁 1 has eigenvalues 푠, . 푠 1 3. The matrices having eigenvalues 푠, are in a single conjugacy class in 푆퐿 푠 2 contained in 푁. 1 푥 1 0 4. , ∈ 푁 0 1 푥 1 5. These matrices generate 푆퐿2.

8 See AMM 116/8 (10/2009), pg. 727 49

A complex is a subgroup of 퐺퐿푛 (ℂ) that is the locus of complex solutions to a finite system of complex polynomial equations in the matrix entries. Cartan’s Theorem: 1. The centers of 푆퐿푛 ℂ , 푆푂푛 ℂ , 푆푃2푛 ℂ are finite cyclic groups. 2. 푆퐿푛 ℂ /푍, 푆푂푛 ℂ /푍, 푆푃2푛 ℂ /푍 are path-connected complex algebraic groups. They are simple except for 푆푂2 ℂ , 푆푂4 ℂ . 3. The other simple, path connected complex algebraic groups, the exceptional groups, are in one of five isomorphism classes. Pf. Based on a classification of Lie algebras.

Exercises 1. Let G be an elementary abelian p-group of order 푛 ≤ 푝. Show that G has no automorphism of order 푝2.

50

11 Representation Theory 11-1 The Basics

Groups are assumed to be finite unless otherwise specified. 푀푛 (퐹) denotes the algebra of 푛 × 푛 matrices over F.

View 1: Representations and Characters  A representation of a group G is a homomorphism from G to the general linear group GL푛 퐹 or GL 푉 (the first sends G to matrices with entries in the field F; the second sends G to linear transformations from the vector space V to itself; they are equivalent since linear transformations correspond to matrices). Two representations 휌: 퐺 → GL 푉 and 휌′ : 퐺 → GL 푉′ are isomorphic if there exists a G-invariant linear transformation 푇: 푉 → 푉′ : ′ 푇 휌푔푣 = 휌푔푇 푣 for all 푔 ∈ 퐺, 푣 ∈ 푉  The character of the representation is the function 휒 푔 = trace 휌푔 .  A vector is G-invariant if 휌푔푣 = 푣 for all 푔 ∈ 퐺, 푣 ∈ 푉, and a subspace W is G- invariant if 휌푔푊 ⊆ 푊 for all 푔 ∈ 퐺 (equality actually holds by invertibility).  A representation 휌: 퐺 → GL 푉 is the direct sum 훼 ⊕ 훽 of two representations 훼, 훽 if there are G-invariant subspaces 푊1, 푊2 such that 푉 = 푊1 ⊕ 푊2 and 훼, 훽 are the projections of 휌 onto 푊1, 푊2. With suitable (change of) basis, all the matrices in the 퐴푔 풪 image of 휌 will be in the form , where 퐴푔 = 훼 푔 , 퐵푔 = 훽(푔). 풪 퐵푔  A reducible representation has a proper G-invariant subspace. (This is equivalent to having nontrivial direct sum decomposition.) If G is a linear group, we simply say G is reducible (the representation being the identity). A character is reducible if its corresponding representation is.

View 2A: Linear Actions and the  A linear action of G on a vector space V over F is a group operation satisfying: o 푔 푣 + 푤 = 푔푣 + 푔푤 for all 푔 ∈ 퐺 and 푣, 푤 ∈ 푉. o 푔 푐푣 = 푐(푔푣) for all 푔 ∈ 퐺, 푐 ∈ 퐹, 푣 ∈ 푉.  푔 ∈ 퐺 operates directly on the vector space, while above, the representation 휌푔 operates on the vector space. Less wordy!  The group ring RG (R a ring) is the set of all linear combinations of elements of G with coefficients in the field R. The elements of G form a basis for the ring. Multiplication is defined like polynomial multiplication:

푎푔푔 푏푔푔 = 푎푔푏푕 푔푕 푔∈퐺 푔∈퐺 푔,푕∈퐺 If R is a field F, then FG is a group algebra.  The group ring acts on V by extending the linear action of G. Thus we say that V is a RG-module. The group ring acts like matrices/ linear transformations acting on the vector space V. The distinct modules correspond to the different representations.

 The character of a FG-module U is the function 휒푈 with 휒푈(푔) defined as the trace of the linear transformation of left multiplication by g. A character is reducible if its corresponding module is simple. 51

Some Algebra Background  A module is… o Simple if it has no proper submodules. o Semisimple if it is the direct sum of simple modules.  An algebra is… o Simple if it has no proper two-sided ideals. (A two sided ideal of A is a subset M so that 푎푀푏 ⊆ 퐴 for any 푎, 푏 ∈ 퐴.) Note that A may be simple as an algebra, but not as an A-module. o Semisimple if all nonzero A-modules are semisimple. Equivalently, A is semisimple as an A-module.  A R-module homomorphism is a homomorphism 휑: 푀 → 푁 such that 휑 푟푚 = 푟휑(푚) for any 푟 ∈ 푅, 푚 ∈ 푀. Hom푅(푀, 푁) is the set of R-module homomorphisms and End푅(푀) is the set of R-module endomorphisms (휑: 푀 → 푀). End푅(푀) forms a division algebra for composition replacing multiplication.  The 푀1 ⊕ 푀2 is the with componentwise addition and scalar multiplication.  The direct sum of algebras 퐴1 ⊕ 퐴2 is the Cartesian product with componentwise addition, scalar multiplication, and multiplication. Note that 푎1 ∈ 퐴1, 푎2 ∈ 퐴2 implies 푎1푎2 = 0. When they are matrix algebras, imagine 퐴1 ⊕ 퐴2 as 퐵1 풪 퐵1 ∈ 퐴1, 퐵2 ∈ 퐴2 . 풪 퐵2

11-2 The Bulk of It

Modules, Linear Actions Representations and Characters G acts linearly on V. The representations of G act on V. Group ring (ex. ℂ퐺) acts on V. The matrices (or linear transformations) in span 휌푔 푔 ∈ 퐺 act on V. R-module homomorphism invariant under Isomorphism of representations invariant R under G. Simple module has no proper submodule. Irreducible representation has no proper G-invariant subspace. A module is the direct sum of simple V is the direct sum of minimal G-invariant submodules. subspaces. This corresponds to a representation being the direct sum of irreducible representations. Schur’s Lemma: If 휑 is a homomorphism Schur’s Lemma: If 휌, 휌′ are irreducible between simple R-modules, then 휑 = 0 or representations on 푉, 푉′, and 푇: 푉 → 푉′ is a 휑 is an isomorphism. G-invariant linear transformation, then Pf. ker 휑 = 0 or 푀. 푇 = 0 or T is an isomorphism. Special case: If S is a simple A-module, If 휌 is an irreducible representation on V, End푅 푆 = 휑 푠 = 푐푠 for some 푐 . then any G-invariant linear operator 푇: 푉 → 푉 is in the form 푇 = 푐퐼. Maschke’s Theorem: Suppose F has Maschke’s Theorem: Suppose F has characteristic 0 or coprime to 퐺 . If U is a characteristic 0 or coprime to 퐺 , and 휌 is a FG-module, and V is a FG-submodule, representation on V. Then V is a direct sum then V is a direct summand of U. Thus, of G-invariant subspaces and 휌 is a direct every FG-module is semisimple (a direct 52 sum of simple modules) sum of irreducible representations Pf. Take an “average”: Write 푈 = 푉 ⊕ 푊, corresponding to these subspaces. and let 휋: 푈 → 푉 be the projection along W. Pf. Take the “average” to get a G-invariant ′ 1 −1 inner product. Take an arbitrary inner Define 휋 푢 = 푔∈퐺 푔휋(푔 푢); then 퐺 product {-,-} and let 〈푣, 푤〉 = 푈 = 푉 ⊕ ker 휋′ as modules. 1 {푔푣, 푔푤}. Then 푉 = 푊 ⊕ 푊⊥ using 퐺 푔∈퐺 the new inner product, with 푊⊥ G-invariant.

F is a trivial FG-module by defining 푔푐 = 푐 The trivial representation is 휌1 푔 = [1] for for every 푔 ∈ 퐺, 푐 ∈ 퐹. all 푔 ∈ 퐺, and the trivial character has

휒1 푔 = 1 for all 푔 ∈ 퐺. If G operates on X, then FX (the set of The permutation representation linear combinations of elements of X) is a corresponding to a G-set X sends each FG-module, called a permutation module. 푔 ∈ 퐺 to the permutation matrix corresponding to left-multiplication by g.

FG is a FG-module. Let 퐺 = 푔푖 1 ≤ 푖 ≤ 퐺 . If 푔푔푖 = 푔휋 푖 , then the 휌reg sends g to the permutation matrix corresponding to π. reg Let A be a semisimple algebra, and Suppose 휌 =⊕ 휌푖 , where each 휌푖 is an suppose that as A-modules 퐴 ≅⊕ 푆푖, where irreducible representation. Then any each 푆푖 is simple. Then any simple A- irreducible representation is isomorphic module S is isomorphic with some 푆 . 푖 with some 휌푖. Pf. Take 0 ≠ 푠 ∈ 푆; define 휑: 퐴 → 푆 by 휑 푎 = 푎푠. The restrictions of 휑 to 푆푖 are 0 or isomorphisms by Schur’s. One of them is an isomorphism. Suppose A is a semisimple algebra. Let M Every representation can be written as a 푟 be an A-module, and write 푀 ≅⊕푖=1 푛푖푆푖, direct sum uniquely: 푟 where 푆푖 are all the distinct simple 휌 =⊕푖=1 푛푖휌푖 modules. Then the 푛푖 are uniquely determined. Pf. Jordan-Hölder for modules. 푛 Let D be a division algebra. Then 푀푛(퐷) is 푀푛 ℂ has no two-sided ideals. ℂ has no a simple algebra and any simple 푀푛(퐷)- 푀푛 ℂ -invariant subsets. Informally, 푀푛 ℂ module is isomorphic with 퐷푛. As modules, can only operate nontrivially on ℂ푛. 푛 푀푛 퐷 ≅ 푛퐷 . The opposite algebra 퐵opp has Transposing matrices, multiplication goes multiplication defined by 푎 ⋅ 푏 = 푏푎. the other way ’round. Wedderburn’s Structure Theorem: An If F is algebraically closed (such as ℂ), then algebra A is semisimple iff it is isomorphic 푛 as algebras, 퐹퐺 ≅⊕푖=1 푀푛푖 퐹 for some 푛푖. with the direct sum of matrix algebras over division algebras. If F is algebraically closed, any semisimple algebra is isomorphic to a direct sum of matrix algebras over F. Pf. 1. If 푆푖 are distinct simple A-modules, and 푟 푈 =⊕푖=1 푛푖푆푖 , then End퐴(푈) ≅ 푟 ⊕푖=1 End퐴 푛푖푆푖 . 53

2. End퐴(푛푖푆푖 ) ≅ 푀푛 (End퐴(푆)) 3. Taking 푈 = 퐴, and taking opposite 푟 op twice, 퐴 ≅⊕푖=1 푀푛푖 End퐴 푆푖 . If A is simple, then A is isomorphic with a Suppose G is an irreducible linear group in 9 matrix algebra over a division algebra. GL푛(퐹). Then span 퐺 = 푀푛(퐹).

11-3 The Character Table and Orthogonality Relations

Basic properties: 1. 휒 1 = dim 휒, the number of rows/ columns in the matrix representation. 2. 휒 is a class function, i.e. a function 퐺 → 퐹 that is constant on every conjugacy class. 3. If g has order k, 휒(푔) is a sum of dim 휒 kth roots of unity. 4. 휒 푔−1 = 휒 ( 푔 ) 5. If 휒, 휒′ correspond to 휌, 휌′, then 휒 + 휒′ corresponds to 휌 ⊕ 휌′. In terms of modules, 휒푈⊕푉 = 휒푈 + 휒푉. 6. 휒푈⊗푉 = 휒푈휒푉 ∗ 7. 휒푈∗ = 휒 푈 , where 푈 = Hom퐹(푈, 푉) (The eigenvalues are inverse.) ∗ 8. 휒Hom 푈,푉 = 휒 푈 휒푉 (Hom 푈, 푉 ≅ 푈 ⊗ 푉)

On irreducible characters: 1. The number of irreducible characters equals the number of conjugacy classes. 푟 a. Suppose ℂ퐺 ≅⊕푖=1 푀푛푖 ℂ is the decomposition into simple algebras. The center Z of ℂ퐺 are all elements with 푔, 푕 having the same coefficients when they are in the same conjugacy class, so has dimension equal to the number of conjugacy classes. Also dimℂ 푍 = 푟 from the decomposition; the center of

푀푛푖 ℂ are scalar matrices. 2. The irreducible characters are linearly independent. 푟 a. Let 푒푖 be the identity element of 푀푛푖 ℂ . If 푗 =1 휆푗 휒푗 = 0 then 푟 0 = 푗=1 휆푗 휒푗 (푒푗 ) = 휆푖 dimℂ 푆푖 so all 휆푖 = 0. 3. Hence, representations (or ℂ퐺-modules) are isomorphic iff they have the same character.

An inner product is defined for characters (and, in general, class functions) as follows: 1 1 〈휒 , 휒 〉 = 휒 푔 휒′ 푔 = 퐶 ⋅ 휒 퐶 휒′ 퐶 푈 푉 퐺 퐺 푔∈퐺 conjugacy class 퐶 For any ℂ퐺-modules U and V, 〈휒푈, 휒푉〉 = dimℂ Homℂ퐺(푈, 푉) Pf. 1 1. Let 푎 = 휒 (푔): Then 푎2 = 푎 so the linear transformation of left multiplication 퐺 푔∈퐺 푈 by a satisfies 푇2 − 푇 = 0 and is diagonalizable with eigenvalues 0 and 1. The 퐺 퐺 eigenspace of 1 is 푈 ; its dimension equals the trace of T. So dimℂ퐺 푈 = 1 휒 (푔). 퐺 푔∈퐺 푈 퐺 2. Homℂ퐺(푈, 푉) = Hom 푈, 푉 ; use (1) and property (8).

Let 휒푖 be the irreducible characters of G. The character table has the characters written

9 푚 The LHS says span 퐺 ≅ 푀푚 (퐹) for some m. Since ℂ is a simple span 퐺 -module, m=n. 54

down the left, and conjugacy classes across the top, with the entries the values the characters take on the conjugacy classes: 퐶1 = {1} ⋯ 퐶푟 휒 푑 1 1 ⋮ ⋮ 휒푟 푑푟 By convention, 휒1 is the trivial character. 0, 푖 ≠ 푗 Row Orthogonality: 〈휒 , 휒 〉 = 훿 = 푖 푗 푖푗 1, 푖 = 푗 Pf. By Schur’s lemma, Homℂ퐺(푆푖, 푆푗 ) = 0 and Homℂ퐺(푆푖, 푆푖) ≅ ℂ. 푟 푟 푟 푟 Cor. 〈 푖=1 푎푖휒푖 , 푖=1 푏푖 휒푖〉 = 푖=1 푎푖 푏푖. If 휒 is a virtual character, 휒 = 푖=1〈휒, 휒푖〉휒푖.

Thus, the characters form an orthonormal basis for the vector space of class functions. The columns are also orthonormal with respect to the same Hermitian form: 0, 푖 ≠ 푗 푟 Column Orthogonality: 푡=1 휒푡 푔푖 휒푡(푔푗 ) = 퐺 = 퐶퐺 푔푖 , 푖 = 푗 퐶푖

Determining the irreducible characters: 1. Find the conjugacy classes of G, to determine the number of irreducible characters. 2. If N is a normal subgroup of G, 휋: 퐺 → 퐺 푁 is the canonical homomorphism, and 휒 is a (irreducible) character of N, then 휒휋 is a (irreducible) character for G. 3. Every irreducible character of a finite abelian group is 1-dimensional. (So any matrix representation is simultaneously diagonalizable.) 4. 푑푖| 퐺 for each i. (Pf. in 12-5) 푟 2 5. 푖=1 푑푖 = |퐺|. 푟 a. By modules: 퐺 = dimℂ ℂ퐺 = 푖=1 dimℂ 푀푑푖 ℂ . reg reg 푟 b. By representations: 〈푥 , 휒푖〉 = dim 휒푖; 퐺 = dim 휒 = 푖=1 푑푖 dim 휒푖 6. Try to find the eigenvalues of representations. For example, if 푥, 푥푎 are conjugate, 푎 then 휌푥 = 휌푥푎 ; if 휆 is an eigenvalue then so is 휆 .

Normal Subgroups 1. Let 퐾휒 = {푔 ∈ 퐺|휒 푔 = 휒(1)}. Then 퐾휒 is a normal subgroup. (The corresponding representation maps 푔 ∈ 퐾휒 to I.) The normal subgroups of G are exactly the sets in

the form 푖∈퐼 퐾휒푖 for some 퐼 ⊆ {1, … , 푟}. 2. Let 푍휒 = 푔 ∈ 퐺 휒 푔 = 휒 1 . If 휒 is irreducible, 푍휒 퐾휒 = 푍(퐺 퐾휒 ). 푍 퐺 = 푟 푖=1 푍휒푖 . If G is nonabelian and simple, 푍휒푖 = 1 for all 푖 > 1.

11-4 Frobenius Reciprocity

Let 퐻 ⊆ 퐺. 퐺  If V is a FH-module, the FG-module Ind퐻 푉 = 퐹퐺 ⊗퐹퐻 푉 is the induced FG-module of V. 퐺 퐺 o If 휙 is the character for V, the induced character of Ind퐻 푉 is denoted by 휙 . 퐺  If U is a FG-module, the FH-module Res퐻 푉 is the restriction of U to H. 퐺 o If 휒 is the character for U, the character of Res퐻 푉 is denoted by 휒 퐻. 퐺 퐺 Note that dim퐹 Ind퐻 푉 = 퐺: 퐻 dim퐹 푉, and as F-vector spaces Ind퐻 푉 =⊕푡∈푇 푡 ⊕ 푉 for T a transversal for H in G. 55

퐺 퐺 Frobenius Reciprocity: As F-vector spaces, Hom퐹퐻 (푉, Res퐻 푈) ≅ Hom퐹퐺(Ind퐻 푉 , 푈). In 퐺 terms of characters, 〈휙, 휒 퐻〉 = 〈휑 , 휒〉. (The left inner product is in H; the right one is in G.)

The Induction-Restriction Table Let 휙푖 1 ≤ 푖 ≤ 푟 , 휒푗 (1 ≤ 푗 ≤ 푠) be the irreducible characters for H and G. The induction- 퐺 restriction table of (G,H) is the 푟 × 푠 matrix with (i,j)-entry equal to 〈휙푖, 휒푗 퐻〉 = 〈휙푖 , 휒푗 〉. The 퐺 ith row gives the multiplicities with which the 휒푗 appear in the decomposition of 휙푖 ; the jth row gives the multiplicities with which the 휙푖 appear in the decomposition of 휒푗 |퐻.

Finding induced characters in G given characters in H: Let 휒 be a character of 퐻 ⊆ 퐺. If no element of H is conjugate to 푔 ∈ 퐺, then 휒퐺 푔 = 0. Else, let s be the number of conjugacy classes of H whose members are conjugate in G to g. Choose representatives of the conjugacy classes 푕푖. Then 푠 푠

퐺 푍퐺 푔 퐶퐻 푕푖 휒 푔 = 휒 푕푖 = 퐺: 퐻 휒 푕푖 푍퐻 푕푖 퐶퐺 푔 푖=1 푖=1 where 퐶퐴 푥 denotes the conjugacy class of x in A and 푍퐴(푥) denotes the centralizer of x in A.

11-5 Applications of Representation Theory

Frobenius’s Theorem: Let G be a transitive permutation group on a finite set X, with each non-identity element fixing at most one element of X. Then the Frobenius kernel 푁= 1 ∪ {푔 ∈ 퐺|푔 has no fixed points} of G is a normal subgroup.

Pf. Let 퐻 = 퐺푥 with irreducible characters 휑푖. Every element of 퐺 − 푁 is conjugate with an element of H; take conjugacy class representatives and use formula in 12-4. Take 휒푖 = 퐺 퐺 푠 휑푖 − 휑푖 1 휑1 + 휑푖 1 휒1 for 1 < 푖 ≤ 푠, and 휒 = 푖=1 휑푖 1 휒푖. Then the Frobenius kernel is 퐾휒 .

Using Algebraic Integers 1. The algebraic integers form a subring of ℂ; that is, the sum and product of algebraic integers is an algebraic integer. A real rational algebraic integer is an integer. 2. For any character 휒 and 푔 ∈ 퐺, 휒 푔 is an algebraic integer. (Sum of roots of unity) 휒 푔 3. For any irreducible character 휒 and 푔 ∈ 퐺, 퐺: 푍 푔 is an algebraic integer. 퐺 휒 1 a. Let S be the corresponding simple module. For C(g) the conjugacy class of g, the map 푆 → 푆 defined by 휑 푠 = 푎푠, 푎 = 푥∈퐶 푔 푥 must be multiplication by a 휒 푔 scalar λ (Schur’s lemma). 휆휒 1 = 푎 = 퐺: 푍 푔 휒(푔) so 휆 = 퐺: 푍 푔 . 퐺 퐺 휒 1 b. 휑 is a linear transformation ℂ퐺 → ℂ퐺. Each entry in the matrix is 0 or 1 so the characteristic equation has integer coefficients. 4. 푑푖| 퐺 for each i. 퐺 |퐺| 휒 퐶 a. = 〈휒, 휒〉 = 퐶 휒′ 퐶 ; each term in the sum is an 휒 1 휒 1 conjugacy class 퐶 휒 1

algebraic integer so 퐺 is a rational algebraic integer and hence an integer. 휒 1 휒 푔 5. If 훾 = is a nonzero-algebraic integer, then 훾 = 1. 휒 1 56

2휋푖 푚 푘 푑 푛 a. 휒(푔) is a sum of roots of unity so 훾 = 푘=1 푒 푘 ∕ 휒 1 for some 푚푘 , 푛푘 . 휒(푔) 2휋푖 푙푘 푑 푛 has conjugates among 푘=1 푒 푘 휒 1 , 1 ≤ 푙푘 ≤ 푛푘, the product of the conjugates must be an integer. But each factor has absolute value less than 1 and not 0, unless 훾 = 1. 6. If G has a conjugacy class of non-trivial prime power, G is not simple. a. Suppose G has a conjugacy class of order 푝푛 containing 푔 ≠ 1. From column orthogonality, 푟 1 휒 푔 휒 1 0 = + 푖 푖 푝 푝 푖=2 휒 푔 휒 1 b. 푖 푖 is not an algebraic integer for some 푖|2 ≤ 푖 ≤ 푟. Then 푝 ∤ 휒 (1); by 푝 푖 Bezout, 푎 퐶: 푍퐺 푔 + 푏휒푖 1 = 1 for integers 푎, 푏. Then multiplying by 휒푖 푔 휒푖 1 , 휒푖 푔 휒푖 푔 = 푎 퐺: 푍퐺 푔 + 푏휒푖(푔) 휒푖 1 휒푖 1

is an algebraic integer by 2 and 3 so 휒푖 푔 = 휒푖 1 , 푔 ∈ 푍휒푖 . 푍휒푖 is nontrivial so G is not simple. (See 12-3)

푎 푏 Burnside’s p-q Theorem: Every group of order 푝 푞 |푎, 푏 ∈ ℕ0, where p, q are distinct primes, is solvable. Pf. Induct on 푎 + 푏. P-groups are solvable. Take Q a q-SSG. For 푄 ≠ 1, 푍 푄 ≠ 1; take 푛 푔 ∈ 푍(푄). Then 퐺: 푍퐺 푔 = 푝 for some n. If n=0 then 푍(퐺) is nontrivial. Else apply 6 to see G is not simple.

Exercises 1. Find the irreducible characters of 퐺 × 퐻 from the irreducible characters of 퐺, 퐻. 2. [PiGT] Let G be a linear group of degree n. a. If G is irreducible and tr 푔 , 푔 ∈ 퐺 takes on at most 푚 values, show that 2 퐺 ≤ 푚푛 . 3 b. If 푔푚 = 1 for all 푔 ∈ 퐺, show that 퐺 ≤ 푚푛 .

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References

Groups and Representations by J. Alperin and Rowen Bell (GTM 162) Algebra by Michael Artin Abstract Algebra: The Basic Graduate Year, by Robert Ash http://www.math.uiuc.edu/~r-ash/ Elements of Abstract Algebra by Richard Dean Problems in Group Theory by John Dixon Permutation Groups by Dixon and Mortimer (GTM 163) The Theory of Groups by Marshall Hall, Jr. An Introduction to the Theory of Groups by Joseph Rotman (GTM 148)

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