Summary

Definitions

Group Theory

Group a set of elements equipped with a binary operation satisfying

• associativity
• identity
• inverse

Subgroup if with closure, identity, and inverse

Abelian Group a group that is commutative, i.e. Cyclic Group a group generated by a single element Normal Subgroup if , , the conjugate Simple Group a group with no proper normal subgroups Commutator Subgroup

Product Group with inherited binary operation Quotient Group group of cosets of

Order , number of elements of a group; smallest number s.t.

Homomorphism where Isomorphism a bijective homomorphism Automorphism an isomorphism from a group onto itself

Kernel Left Coset , , Index for , the number of cosets of contained in

Group Actions

Group Action given a group and set , where

Orbit acting on , , Stabilizer acting on , ,

Transitive An action with only one orbit Faithful An action where IFF

Centralizer Conjugacy Class Normalizer

p-group A group where for some prime

Representation Theory

Complex Representation a homomorphism

Character trace of a representation Inner product

Dimension the dimension of the codomain of , or G-invariant a vector or space fixed by for all ; a linear operator compatible with , Isomorphic two irreps and are isomorphic if there is a G-invariant linear map , Irreducible A representation with no G-invariant subspaces

Permutation Representation Let act on , and be the permutation matrix. This is a representation of a vector space with basis . Regular Representation is the permutation representation of acting on itself by left multiplication

Useful Tools

Chinese Remainder Theorem For , where are pairwise coprime,

Lagrange’s Theorem For , divides

Correspondence Theorem Let be a surjective homomorphism with kernel . Then is a bijection between subgroups of and subgroups of containing .

Direct Product Theorem Let , and .

• is injective IFF
• is a homomorphism IFF for all , ,
• If normal, then
• is an isomorphism IFF , , and

First Isomorphism Theorem Let be surjective with kernel . is isomorphic to , and there is a unique isomorphism s.t. , where is the canonical quotient map.

Orbit Stabilizer Theorem Let act on and denote . For , there is a bijection with

Counting Formula

Class Equation

Conjugating Permutations For ,

First Sylow Theorem A finite group with order divisible by prime has a Sylow p-subgroup

Second Sylow Theorem Let be a finite group with order divisible by prime

• The Sylow p-subgroups are conjugate subgroups
• Every subgroup of that is a -subgroup is contained in a Sylow p-subgroup

Third Sylow Theorem Let be a finite group with order divisible by prime . and is the number of Sylow p-subgroups. Then divides and

Classification of Finite Abelian Groups Every finite abelian group is isomorphic to direct product of cyclic groups. These are isomorphic up to factorization by Chinese Remainder Theorem

Standard Representation For a finite rotation group, this is the representation on , where represents rotations on this space

Maschke’s Theorem Every finite representation on a non-zero, finite-dimensional complex vector space is the direct sum of irreducible representations.

Main Theorem of Characters Let be a finite group

• The irreducible characters of are orthonormal
• There are as many isomorphism classes of irreps as conjugacy classes of the group
• Let be the irreps of , and their characters. Then divides and

Characters of Finite Abelian Groups Let be a finite abelian group

• Every irrep of has dimension 1. The number of irreps is
• Every matrix representation of is diagonalizable

Schur’s Lemma

• Let and be irreps of to and . Let be G-invariant. is an isomorphism, or
• Let be an irrep of on , and let be G-invariant. Then is multiplication by a scalar,

Useful Groups

• symmetric group permutations
• alternating group even permutations
• cyclic groups symmetries of n-gon w/o inversions
• dihedral groups symmetries of n-gon with inversions
• Klein 4 group smallest non-cyclic simple group
• quaternion group , ,
• tetrahderal group symmetries of tetrahedron w/o inversions
• octahedral group symmetries of octahedron w/o inversions
• icosahedral group symmetries of icosahedron w/o inversions
• invertible matrices over a field
• invertible matrices over a vector space
• determinant 1 matrices over a field
• additive group of integers modulo
• multiplicative group of integers modulo prime

Magic Tricks

Maps

• A homomorphism is injective IFF
• The kernel of a homomorphism is a normal subgroup
• There is a homomorphism , where permutes elements in partitions of , while permutes these partitions.
• Left multiplication and conjugation are automorphisms
• Inversion is an automorphism IFF the group is abelian
• If is simple, then every homomorphism has kernel or
• If is prime, then (multiplication group )

Groups

• If is prime, then is cyclic
• The order of divides
• Work with groups, subgroups, or cosets instead of element chasing
• , then for ,
• , then

More Groups

• Abuse counting formula
• Let and . and IFF
• iff the elements of commute with the elements of
• then the commutator ,

Isomorphism Classes

• Compare group order
• Compare centralizer
• Apply product theorem (only for proof, not disproof)

Group Actions

• Consider group acting on itself, or its cosets by conjugation or left multiplication
• Construct homomorphism from aforementioned group action to
• Look at orbits or stabilizers of elements.

Sylow

• If a group has only one Sylow-p subgroup, that subgroup is normal
• Do casework on normality
• The center of a -group is nontrivial
• implies is abelian
• cyclic implies is abelian
• For , is a simple group
• Let be a Sylow -subgroup. Then the index of is
• Consider group acting on subgroup by conjugation to determine commutativity

Rep Theory

• For , for all irreps of .
• For , for all irreps of .
• For any irrep ,
• If , then for irrep of , is an irrep of .
• Quotient out and compute char table of .
  • Reindex group summation
• Average linear transform . If is G-invariant, then
• The columns and rows of a character table are orthogonal
• Scaling the columns of a character table by yields a unitary matrix
• The character of a permutation matrix is the number of elements it fixes.
• Take inner product of a character with itself, or other irreps to find its composition
  • IFF is irreducible
  • IFF is sum of two irreducible characters
  • IFF is sum of three irreducible characters
  • IFF is sum of four irreducible characters or two copies of one irreducible character
• , where is dimension of the irrep
• Square group summation and reindex
• Apply trace to averaged function