Algebra 3

Coset

Let G be a group and H a subgroup. Let g be an element of G.

• The set gH = {gh : h ∈ H} is a left _ of H in G

• The set Hg = {hg : h ∈ H} a right _ of H in G

When is g1H = g2H?

if and only if g₁^-1g₂∈

Index

Let G be a group and H be a subgroup.

• The left __of H in G, denoted by [G : H], is the number of left cosets of H in G.

• The right __of H in G is the number of right cosets of H in G.

When does #gH = #H

Let G be a group and H a finite subgroup. If g ∈ G then gH and Hg

Construct a bijection between H and gH

Partition

Let A be a set and B1, B2, . . . , Bn be subsets of A.

•  A = B1 ∪ B2 ∪ · · · ∪ Bn

•  if i ≠ j then Bi ∩ Bj = ∅ (i.e. the Bi are pairwise disjoint)

• Bi ≠ ∅ for i = 1, . . . , n

Lagrange’s Theorem

Let G be a finite group and H a subgroup.

#G = [G : H] · #H

Conjugate

Let H be a subgroup of G.

A __of H has the form gHg^-1 for some g ∈ G.

Normal subgroup

A subgroup H of G is __if and only if gHg^-1 = H for all g ∈ G

That is H is equal to all its conjugates. We write H ⊴ G.

Quotient

Let G be a group and H a subgroup.

Write G/H for the set of left cosets of H in G: G/H = {gH : g ∈ G}.

Multiplication of G/N

(gN)(g′N) = gg′N

Quotient Group of G over N

Let G be a group and N a normal subgroup. Then G/N with its operation is a group with identity element N = 1_GN and inverses given by (gN)⁻¹ = g⁻¹N

If G is finite then #(G/N) = [G : N] = #G #N .

Linear Map

Let F be a field. Let V and W be F-vector spaces.

A map ϕ : V → W satisfying the two conditions

• ϕ(v1 + v2) = ϕ(v1) + ϕ(v2) for all v1, v2 ∈ V

• ϕ(λv) = λϕ(v) for all λ ∈ F and v ∈ V

Kernel and Image

Ker(ϕ) = {v ∈ V : ϕ(v) = 0} - subspace of V

Im(ϕ) = {ϕ(v) : v ∈ V } - subspace of W

Homomorphism

Let G, H be groups and let ϕ : G → H be a map.

ϕ is a __ of groups if ϕ(gh) = ϕ(g)ϕ(h) for all g, h ∈ G.

Isomorphism

Let G and H be groups.

A map ϕ:G → H is an __ if it is a bijective homomorphism (G ∼= H)

What is Ker(ϕ) and Im(ϕ)?

• Ker(ϕ) is a normal subgroup of G

• Im(ϕ) is a subgroup of H.

The First Isomorphism Theorem

Let ϕ : G → H be a homomorphism of groups.

Let ϕˆ : G/ Ker(ϕ) → Im(ϕ),

ϕˆ(g Ker(ϕ)) = ϕ(g)

Then ϕˆ is a well-defined group isomorphism.

Even/ Odd permutation

• can be written as a product of an even number of transpositions

• can be written as a product of an odd number of transpositions

Sign

σ ∈ Sn by sign(σ) = ( 1 if σ is even −1 if σ is odd)

sign(σ) = (-1)^m for σ = ε1,ε2,…,εm where εi are transpositions

It is a homomorphism

What is An in relation to Sn /what is its size?

• Let n ≥ 2. Then An is a normal subgroup of Sn.

• [Sn : An] = 2

• #An = #Sn/2 = n!/2

Isomorphic

Let G and H be cyclic groups of order n. Then G and H are _________.

Let G be a cyclic group of order n, then G≅Cn (cyclic group of order n)

Prime

Let p be a ___. Any group of order p is isomorphic to Cp.

Direct Product

• Let G and H be groups. The ____ of G and H is: G × H = {(g, h) : g ∈ G, h ∈ H}

• i.e. G × H is the set of ordered pairs (g, h) where g ∈ G and h ∈ H

• The binary operation on G × H is (g1, h1) · (g2, h2) = (g1g2, h1h2)

Groups of order 4

C₄

C₂xC₂

Fundamental Theorem of Finite Abelian Groups

Let G be a non-trivial finite abelian group. Then there are integers b1 | b2 | · · · | bn, all > 1, such that G ≅ C_b1 × C_b2 × · · · × C_bn .

#G = b1b2 · · · bn.

Dihedral group

Let n ≥ 3. The regular n-gon has 2n symmetries which consist of n rotations and n reflections which form a group D2n.

• Centered at the origin, with one vertex lying on the x-axis, which is labelled as 1.

• The numbering goes anticlockwise.

• The symmetry r is anticlockwise rotation around the centre through angle 2π/n.

• The symmetry s is reflection in the x-axis

D2n and #D2n

Let n ≥ 3. Then D2n = {id, r, r2 , . . . , rn−1 } ∪ {s, sr, sr2 , . . . , srn−1 }.

#D2n = 2n and R is a normal subgroup of index 2.

a = id and b=c

• Let a ∈ D2n. Suppose a fixes vertices 1 and 2.

• Let b, c ∈ D2n. Suppose b(1) = c(1) and b(2) = c(2).

rⁿ, s², srs = ??? 

With r, s as above, rⁿ = id, s² = id, srs = r^-1

Word

• Let G be a group. Let g1, . . . , gn be elements of G. A ___ in g1, g2, . . . , gn is a finite product of g₁, g⁻¹₁ , g₂, g⁻¹₂ , . . . , g_n, g⁻¹_n .

• For example, if g, h ∈ G, then the following are words in g, h: h³, g⁻¹h, h⁻²g⁻¹h⁻¹g³h, 1_G.

The subset of words is a _____

Let G be a group. Let g1, . . . , gn be elements of G. Write ⟨g1, . . . , gn⟩ for the subset of words in g1, . . . , gn. Then ⟨g1, . . . , gn⟩ is a subgroup of G

Group Presentation

To specify a group by specifying generators and relations.

The notation has the form G = ⟨S|R⟩ where S is a set of symbols, and R is a set of relations between the symbols.

Fundamental Theorem of Group Presentations

Let G = ⟨S | R⟩ be a group presentation where S = {s1, s2, . . . , sn} is a finite set of generators and R is a set of relations. Let H be a group and let h1, h2, . . . , hn be elements of H.

There exists a homomorphism ϕ : G → H satisfying ϕ(si) = hi if and only if every relation r ∈ R holds with the si replaced by the hi.

Moreover, in this case the homomorphism ϕ is unique.

Quaternion group

Q8 = ⟨a, b | a⁴ = id, a² = b² , bab⁻¹ = a⁻¹⟩.

Elements of Q8

Every element of Q8 can uniquely be written as aⁱ b^j with 0 ≤ i ≤ 3 and 0 ≤ j ≤ 1. Thus #Q8 = 8

Exponent

Let n be a positive integer. We say that a group G has ____ n if n is the smallest positive integer such that gⁿ = id for all g ∈ G

Abelian

Let G be a group with exponent 2.

• Then G is ____.

• Then, for some positive integer n, G≅Cⁿ₂ . In particular, #G = 2ⁿ.

Isomorphism of groups order 6

Let G be a group of order 6. Then G≅C6 or G≅D6.

Isomorphism of groups order 8

Let G be a group of order 8. Then G is isomorphic to one of

• C₂×C₂×C₂

• C₂×C₄

• C₈

• D₈

• Q₈

Left Action

Let G be a group and X a set. A ____ of G on X is a map G × X → X, (g, x)→g ∗ x which satisfies the following two properties:

• (A1) 1_G∗x = x for all x ∈ X

• (A2) (gh)∗x = g∗(h∗x) for all g, h ∈ G and x ∈ X

∼ an equivalence relation

Let G act on X. For x, y ∈ X write x ∼ y if and only if there some g ∈ G such that g ∗ x = y. Then ∼ is an _____.

Orbit

We define the ___ of x ∈ X under the action of G by Orb_G(x) = {g∗x : g ∈ G}.

They form a partition of X.

Acts transitively

G acts ___ on X if, for any x, y ∈ X, there is some g ∈ G such that g ∗ x = y. ↔ Orb_G(x) = X 

Fixed

We say that x ∈ X is _____ by G if g ∗ x = x for all g ∈ G.

We write Fix(G) for the set of x ∈ X that are ___ by G.

Observe that x ∈ Fix(G) ⇐⇒ x is fixed by G ⇐⇒ OrbG(x) = {x}.

Stabilizer

Suppose G acts on X. The ___ of x ∈ X is Stab_G(x) = {g ∈ G : g ∗ x = x} the set of elements of G that fix x. It is a subgroup of G.

Orbit-Stabilizer Theorem

Let G be a finite group acting on a finite set X, and let x ∈ X.

Then #G = # Orb_G(x) × # Stab_G(x), or equivalently [G : Stab_G(x)] = # Orb_G(x).

p be a prime and let X be a finite set. Suppose Cp acts on X.

(i) Every orbit has size 1 or p.

(ii) # Fix(Cp) ≡ #X (mod p).

(#G)^p-1

Let G be a finite group and p a prime. Then

____ ≡ (#{g ∈ G : g p = id}) (mod p).

Cauchy’s Theorem

Let G be a finite group and let p be a prime. Suppose p | #G. Then G has an element of order p

Conjugate

Let G be a group. Two elements h1, h2 ∈ G are ____ if there is some g ∈ G such that gh₁g⁻¹ = h₂.

conjugation

The group G acts on itself by _____ G × G → G, g ∗ h = ghg⁻¹ .

Conjugacy class

of h ∈ G is simply the orbit of h under the conjugation action of G:

Cl_G(h) = {ghg⁻¹ : g ∈ G}

Centralizer

of h ∈ G is simply the stabilizer of h: C_G(h) = {g ∈ G : ghg⁻¹ = h} = {g ∈ G gh = hg}.

Cycle type

Let σ ∈ Sn be a permutation. We say that σ has ___ 1^r1 2^r2 3^r3 4^r4 · · · if its disjoint cycle decomposition has exactly r1 cycles of length 1, r2 cycles of length 2, r3 cycles of length 3, and so on.

aba⁻¹

Let a = (x1, x2, . . . , xr) be an r-cycle in Sn. Let b ∈ Sn. Then aba⁻¹ = (b (x1), b(x2), . . . , b(xr))

Two permutations are conjugate

in Sn are ___ if and only if they have the same cycle type.

Unit

Let R be a ring. An element u is called a ____ if there is some element v in R such that uv = vu = 1.

Field

a commutative ring in which every non-zero element has a multiplicative inverse. (every non-zero element is a unit)

Unit group

of R to be the set R^∗ = {a ∈ R : a is a unit in R}.

Norm map

N: Z[i] → Z by N(a + bi) = a² + b² , a, b ∈ Z.

Multiplicative

Let α, β ∈ Z[i]. Then N(αβ) = N(α)N(β).

Unit group of Z[i]

{1, −1, i, −i}

Ring homomorphism

Let R, S be rings. A ____ ϕ : R → S is a function that satisfies

• ϕ(a + b) = ϕ(a) + ϕ(b) for all a, b ∈ R

• (b) ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ R

• (c) ϕ(1R) = 1S

Kernel

Let ψ : R → S be a homomorphism of rings.

Ker(ψ) = {r ∈ R : ψ(r) = 0}.

Image

Im(ψ) = {ψ(r) : r ∈ R}.

First Isomorphism Theorem

Let ψ : R → S be a homomorphism of rings.

• Ker(ψ) is an ideal of R

• Im(ψ) is a subring of S

• The induced map ψˆ : R/ Ker(ψ) → Im(ψ), ψˆ(r + Ker(ψ)) = ψ(r) is an isomorphism.

Zero divisor

Let R be a commutative ring. An element x ≠ 0 is called a ____ if there is y ≠ 0 in R such that xy = 0.

Integral domian

An ___ is a non-zero commutative ring that has no zero divisors. Thus if xy = 0 then x = 0 or y = 0.

Field and subring of a field

Any ___ is an integral domain. A ___ is an integral domain.

Cancellation law for integral domain

Let R be an integral domain. Let a, b, c ∈ R with a ≠ 0. If ab = ac then b = c.

Finite

Every ___ integral domain is a field.

Divides

Let R be an integral domain. Let a, b ∈ R. We say that a ___ b and write a|b if b = ac for some c ∈ R.

Associates

We say that a, b are ____ (and write a ∼ b) if a | b and b | a.

Proper divisor

We say that a is a ___ of b if a | b and b ∤ a; this is equivalent to a | b and a ̸∼ b.

When does a~b

Let R be an integral domain and let a, b ∈ R.

____ if and only if there is a unit u ∈ R∗ such that a = ub.

Irreducible

Let R be an integral domain. An element a ∈ R is called _____if it satisfies the following:

• a ≠ 0

• a is not a unit of R

• if a = bc with b, c ∈ R, then either b is a unit or c is a unit

Prime

An element π ∈ R is called ____ if it satisfies the following

• π ≠ 0

• π is not a unit of R

• if π | bc with b, c ∈ R, then π | b or π | c.

Every prime

Let R be an integral domain. Then __ of R is irreducible.

Unique Factorisation Domain

Let R be an integral domain. We say that R is a UF if

• (existence of factorisation) every non-zero element a can be written in the form a = up1p2 · · · pr where the pi are irreducible and u is a unit

• (uniqueness of factorisation) whenever up1p2 · · · pr = vq1q2 · · · qs with pi , qj irreducible, and u, v are units, then r = s and we can reorder the qj so that pi ∼ qi for i = 1, 2, . . . , r.

Division with remainder for integers

Let m, n ∈ Z with n ≠ 0. Then there are unique q, r ∈ Z such that m = qn + r, 0 ≤ r < |n|.

We call q the quotient and r the remainder obtained upon dividing m by n.

Division with remainder for polynomials

Let F be a field. Let g, f ∈ F[X] with f ≠ 0. Then there are unique q, r ∈ F[X] with g = qf + r, r = 0 or deg(r) < deg(f).

We call q the quotient and r the remainder obtained upon dividing g by f.

Euclidian function

Let R be an integral domain. A ___ on R is a map ∂ : R \ {0} → N satisfying the following two conditions:

• if a, b ∈ R \ {0} and a | b then ∂(a) ≤ ∂(b)

• if a, b ∈ R, and b ≠ 0, then there exists q, r ∈ R such that a = bq + r with either r = 0 or ∂(r) < ∂(b).

Euclidian ring/ domain

A pair (R, ∂) where R is an integral domain, and ∂ is a Euclidean function on R.

F[X] and deg

Let F be a field. Then __ is a Euclidean domain with Euclidean function _.

R is a Euclidian domain.

Let R be a ____. Then R is a unique factorization domain.

a~b

Let R be a Euclidean domain. Let a, b be non-zero elements of R with a|b and ∂(a) = ∂(b). Then ____.

In particular, if a is a proper divisor of b, then ∂(a) < ∂(b).

Product of irreducibles

Let R be a Euclidean domain. Every element that is neither 0 nor a unit is a ___

Coprime

Let R be an integral domain, and let a, b ∈ R. We say that a, b are ___, if the only elements of R that divide both a and b are units.

a is irreducible

Let R be an integral domain. Let a, b ∈ R and suppose ____. Then either a | b or a and b are coprime.

Bezout’s theorem

Let R be a Euclidean domain. Let a, b ∈ R be coprime. Then there are x, y ∈ R such that xa+yb = 1.

the same

Let R be a Euclidean domain. Then in R the primes and irreducibles are ___.

Allowable

Let R be a Euclidean ring and let A ∈ Mm,n(R). Write Ri for the i-th row and Ci for the i-th column. We call the following elementary row and column operations ____:

• Ri ↔ Rj : swap the i-th and j-th rows (i ≠ j).

• Ci ↔ Cj : swap the i-th and j-th columns (i≠ j)

• Ri → Ri+qRj : add q times the j-th row to the i-th row (i ≠ j, q ∈ R)

• Ci → Ci+qCj : add q times the j-th column to the i-th column (i ≠ j, q ∈ R)

• Ri → uRi : multiply the i-th row by unit u ∈ R^∗

• Ci → uCi : multiply the i-th column by unit u ∈ R^∗

Smith Normal Form

Let R be a Euclidean domain. Let A ∈ Mm,n(R). Then there is a finite sequence of allowable elementary operations such that A becomes of the form (X.26)

where the bi are non-zero elements of R and b1 | b2 | · · · | br

UAV

Let R be a Euclidean ring. Let A ∈ Mn(R). Then there are matrices U, V ∈ GLn(R) such that ___ is in Smith Normal Form.

R-module

an additive abelian group (M, +, 0) equipped with an operation

R × M → M, (r, m) → rm (scalar multiplication)

that satisfies the following properties:

• 1 · m = m for all m ∈ M

• (r · s) · m = r · (s · m) for all r, s ∈ R and m ∈ M

• (r + s) · m = r · m + s · m for all r, s ∈ R and m ∈ M

• r · (m + n) = r · m + r · n for all r ∈ R and m, n ∈ M.

R-submodule

of M is a subgroup (N, +, 0) of (M, +, 0) that satisfies r · n ∈ N for all r ∈ R and n ∈ N.

Quotient module

M/N to be the set of cosets m + N with m ∈ M. Addition and scalar multiplication are given in the natural way

(m1 + N) + (m2 + N) = (m1 + m2) + N

r · (m + N) = rm + N

Homomorphism of R-modules

Let M, N be R-modules. A map ϕ : M → N is a _____ if

• ϕ(m1 + m2) = ϕ(m1) + ϕ(m2)

• ϕ(rm) = rϕ(m).

First Isomorphism Theorem (R-modules)

Let ϕ : M → N be a homomorphism of R-modules.

• Ker(ϕ) is an R-submodule of M

• Im(ϕ) is an R-submodule of N

• The induced map ϕˆ : M/ Ker(ϕ) → Im(ϕ), ϕˆ(m + Ker(ϕ)) = ϕ(m) is an isomorphism of R-modules.

R-span

Let M be an R-module. Let X = {x1, . . . , xn} be a finite subset of M. We define the __ of X to be SpanR(X) = {r1x1 + · · · + rnxn : ri ∈ R} .

This the set of all linear combinations of elements of X with coefficients in R.