Math 411 Final Exam

action of a group on a set, G ↷ X

it’s a homomorphism G → Sym(X)

orbit of a element under an action, G · x or O(x)

this is the subset of S containing all points reachable from x by applying any g ∈ G; {gx | g ∈ G}

orbit space, X/G

{Gx : x ∈ X}

transitive action

type of action where G ↷ X is transitive, meaning that the action can move any x ∈ X to any other element x’ ∈ X; in this case, there is only one orbit

Stab(x)

{g ∈ G : gx = x}, subgroup of G

fixed points of a group element, X^g

set of fixed points in a set X under the action of a group G; {x ∈ X : gx = x}

Automorphism group , Aut(G)

a function Φ: G → G that is a bijection and a group homomorphism

Center of a group, Z(G)

set of elements that commute with every element in the group; {z ∈ G : zg = gz}

Inner automorphisms, Inn(G)

a normal subgroup of an automorphism that maps a group to itself while preserving its structure

Orbit-stabilizer theorem

Given G ↷ X with finite G, |O(x)| = [G : Stab(x)] = |G| / |Stab(x)|

Burnside’s lemma

If G ↷ X and both G and X are finite, then |X/G| = 1/|G| * Σ|X^g| for all g ∈ G

Aut(Z)

{+1, -1}

Aut(Z/nZ)

(Z/nZ)^x, the elements with multiplicative inverses

Aut(Z/2Z x Z/2Z)

isomorphic to S₃ and GL₂(Z/2Z)

Cauchy’s theorem

if p divides |G|, then G has element of order p

First Sylow Theorem

Let G be finite group. For prime number p, if p^k divides |G|, there is subgroup H of G with |H| = p^k

kernel of a homomorphism

given f: G → H, this is the set of elements in the domain G that map to the identity element e_H in the codomain H; {x ∈ G : f(x) = e_H}

isomorphism

a homomorphism that is also a bijection (one-to-one and onto, or injective and surjective)

sign of a permutation, sgn(σ)

+1 for even number of permutations, -1 for odd number of permutations

left coset xH

Let subgroup H < G, this is equal to {xh : x ∈ G and for all h ∈ H}

right coset Hx

Let subgroup H < G, this is equal to {hx : x ∈ G and for all h ∈ H}

quotient group

For a normal subgroup N of G, this group G/N is the set of all cosets {gN : g ∈ G}

equivalence relation

a binary relation that connects elements that are reflexive, symmetric, and transitive

partition of a set X

a grouping of elements in X into non-empty, disjoint subsets such that every element in X is in exactly one of these subsets

set of left H-cosets, G/H

these cosets form a partition of group G, so they are disjoint and their union is G

semidirect product

a group G with a normal subgroup N and a subgroup C such that N and C are disjoint and under the quotient map G → G/N, we get an isomorphism

Span_Z(S)

For a subset S = {x₁, …, x_n}, its ____ is {m₁x₁ + m₂x₂ + … + m_nx_n : m_j ∈ Z}; this is equal to <S>

free abelian group

an abelian group with a basis, meaning it is isomorphic to a direct sum of copies of the integers Z

basis

a linearly independent set of generators for some abelian group A

torsion abelian group

an abelian group where every element has finite order

rank

given a finitely generated abelian group A, _____ is the cardinality of A

Symmetric group of a set X, Sym(X) or S_X

group of all bijections from set to itself using function composition as operation

Symmetric group of {1, 2, …, n}, S_n

This group represents all possible permutations of the n! elements in this group

Alternating group A_n

also known as the kernel ker(sgn); {σ ∈ S_n : sgn(σ) = 1} = {even presentations in S_n}

Dihedral group D_n

this group is made up of the geometric symmetries of the regular n-gon (n >= 3) with n reflective symmetries and rotations of r 360/n degrees; < r, s | rⁿ = e, s² = e, sr = r^-1s >, has order 2n

Cyclic group C_n or Z/nZ

<r | rⁿ = e>, has order n

Infinite cyclic group

the set of integers Z form this type of group under addition, (Z, +)

2d rotation group SO(2)

General Linear Group GL_n(R)

{A ∈ M_nxn(R) : det(A) ≠ 0}

SL_n(R)

{A ∈ GL_n(R) : det(A) = 1}

Orthogonal group, O(n)

{A ∈ GL_n(R) : A^TA = 1_n}

Rect, or Z/nZ x Z/nZ

<h, v | h² = e, v² = e, hv = vh>

D_n as a semidirect product of Z/nZ ⋊ Z/nZ

D_n is built from <r> and <s> but is not a direct product

Smith Normal Form Theorem

any n x m integer matrix A ∈ Mat_nxm(Z) is equivalent to an integral matrix of this form (d₁, d₂, …, d_k along the diagonal) where 0 < d₁ <= d₂ <= … <= d_k and d₁ | d₂, d₂ | d₃, …, d_k-1|d_k

Cancellation property

let a, x, y ∈ G. If ax = ay, then x = y because a^-1ax = a^-1ay → x = y

Left multiplication function L_g

L_g is a bijection G→G, and is therefore invertible

Cayley’s theorem

every group is a subgroup of a symmetric group. If the order of G is n, then G is a subset of S_n

cyclic

Subgroup <g> generated by element g is ___________

abelian

if x and y in G commute, the subgroup <x,y> is ______ and a product of cyclic groups

identity, inverse, associability

A nonempty subset is a subgroup if and only if it satisfies the _____, ______, and ______ properties

image

under a homomorphism, the ______ of a subgroup is a subgroup

preimage

under a homomorphism, the ______ of a subgroup is a subgroup

normal subgroup

given H < G, H is a ______________ if either xH= Hx, xhx^-1 ∈ H for any x∈G and h∈H, or xHx^-1 for all x ∈ G

bijective

Every left coset xH is _________ to H via L_x

equivalence relation

Being in the same coset is an ___________________________

Lagrange’s theorem

if G is a finite group, then |H| | |G| for any subgroup H. In fact, |G|/|H| = number of left H-cosets in G = [G:H] = index of H in G

divides

For a finite group, the order of a group element ________ the order of the group

Z/pZ

A group G with prime order p is isomorphic to _________

Noether 1st isomorphism theorem

for any group map Φ : G→H, there is a group isomorphism Φ^~: G/ker(Φ) → image(Φ) such that Φ = Φ^~ * q_ker(Φ); in other words, if Φ : G→H is a group homomorphism, then the quotient group G/ker(Φ) is isomorphic to image im(Φ) = Φ(G)

(A/H) x (B/K)

If H<A and K<B are all abelia, then (A x B)/(H x K) = __________

dZ

Every subgroup of Z is in form _____ for some natural number d

cyclic

every subgroup of Z is _______

cardinality

Any basis of finitely generated free abelian group has the same ________

at most n

Every subgroup of Zⁿ is free and finitely generated with rank ________

corollary of Smith Normal Form

every subgroup of Zⁿ is isomorphic to a subgroup d₁Z x d₂Z x … x d_kZ x 0Z x … x 0Z

basis {v₁, …, v_n}

For any subgroup H < Zⁿ, there is a ______________ of Zⁿ such that H = span_Z(d₁v₁, …, d_kv_k) with 0 < d₁ < … < d_k and d₁ | d₂, d₂ | d₃, …, d_k-1 | d_k

Fundamental theorem in invariant form

given finitely generated abelian group A, A is isomorphic to Z/d₁ x Z/d₂ x … x Z/d_k x Z^r where r = rank(A) for natural numbers r >= 0 and 0 < d₁ < d₂ < … < d_k and d₁ | d₂, d₂ | d₃, …, d_k-1 | d_k

Fundamental theorem in elementary divisor form

if A is a finitely generated abelian group with |A| = n, then A is isomorphic to Z/p₁^k₁₁ x … x Z/p₁^k_1i x Z/p₂^k₂₁ x … x Z/p₂^k_{2_i} x Z/p_c^k_{ci_c} where n = p₁^{k₁₁+…+k_1i} + … + p_c^{Σk_{c_j} for j = 1 to i_c}