Intro to Abstract Algebra // Flashcards
Binary operation
A binary operation on a set S is a rule which for every two elements of S gives another element of S
Commutative
A binary operation ◦ is commutative on S if a ◦ b = b ◦ a for all a, b ∈ S
Associative
A binary operation ◦ is associative on S if (a ◦ b) ◦ c = a ◦ (b ◦ c) for all a, b, c ∈ S
9 axioms of the real numbers
Addition:
- is commutative
- is associative
- has an additive identity element (0)
- each a ∈ ℝ has an additive inverse, -a
Multiplication:
- is commutative
- is associative
- has a multiplicative identity element (1)
- each a ∈ ℝ/{0} has a multiplicative inverse, 1/a
The ninth is that multiplication distributes over addition
Composition
If U, V, W are three sets, and f,g are functions such that f: U → V, g: V → W, then we define the composition g ◦ f: U → W by the rule (g ◦ f)(x) = g(f(x))
Group (G, ◦)
A group is a pair (G, ◦) where G is a set and ◦ is a binary operation on G, such that the following hold:
(i) (closure) for all a, b ∈ G, a ◦ b ∈ G;
(ii) (associativity) for all a, b, c ∈ G,
a ◦ (b ◦ c) = (a ◦ b) ◦ c;
(iii) (existence of an identity element) there is an element e ∈ G such that for all a ∈ G,
a ◦ e = e ◦ a = a;
(iv) (existence of inverses) for every a ∈ G, there is an element b ∈ G such that
a ◦ b = b ◦ a = e.
Abelian group
A group (G, ◦) is abelian if, in addition to the first four properties, it is also commutative
Order of an element
The order of an element a in a group G is the smallest positive integer n such that a^n = 1. If there is no such positive integer n, we say a has infinite order
Order of a group
The order of a group G is the number of elements that G has, denoted by |G| or #G
Subgroup H of G
Let (G, ◦) be a group. Let H be a subset G, then if (H, ◦) is also a group, H is a subgroup of G
Lagrange’s theorem v1
Let G be a finite group, and let g be an element of G. Then the order of g divides the order of G
Lagrange’s theorem v2
Let G be a finite group, and H a subgroup of G. Then the order of H divides the order of G
Cyclic subgroup ⟨g⟩
The subgroup ⟨g⟩ = {g^n : g ∈ ℤ} is called the cyclic subgroup generated by g
Isomorphism
Let (G, ◦) and (H, ⁕) be groups. We say that the function φ: G → H is an isomorphism if it is a bijection and it satisfies φ(g1 ◦ g2) = φ(g1) ⁕ φ(g2) for all g1, g2 in G. (G and H are isomorphic)
Left coset of H in G
Let G be a group and H a subgroup. Let g ∈ G. We call the set gH = {gh : h ∈ H} a left coset of H in G
Right coset of H in G
Let G be a group and H a subgroup. Let g ∈ G. We call the set Hg = {hg : h ∈ H} a right coset of H in G
Index of H in G
Let G be a group and H a subgroup. The index of H in G is the number of left closets of H in G, denoted by [G : H]
Lagrange’s theorem v3
Let G be a finite group and H a subgroup. Then |G| = [G : H] * |H|
Congruent modulo H
Let (G, +) be an abelian group, where the binary operation is addition, and let H be a subgroup. Let a, b ∈ G. a, b are congruent modulo H if a - b ∈ H, written a ≡ b (mod H)
Congruence class
Let (G, +) be an additive abelian group and H a subgroup. Let a ∈ G. We shall denote by ā the congruence class of a modulo H, defined by ā = {b ∈ G : b ≡ a (mod H)}
Injective
f is injective if whenever a1, a2 ∈ A, a1 ≠ a2 then f(a1) ≠ f(a2)
Surjective
f: A → B is surjective if for every b ∈ B, there is some element a ∈ A such that f(a) = b
Pigeonhole principle
Let A be a finite set and let f be a function from A to A. Then f is injective ⇔ f is surjective
Identify map on A
The identity map on a set A is the map idA : A → A satisfying idA(x) = x for all x ∈ A
Invertible
Let f : A → A be a function on A. We say that f is invertible if there exists a function g : A → A such that f ◦ g = g ◦ f = idA. We call g the inverse of f and denote it f^-1
n-th symmetric group
The group Sym({1,2,…,n}), denoted Sn
Ring (9 properties)
A ring is a triple (R, +, •) where R is a set and +, • are binary operations on R such that the following properties hold:
1-5 (R, +) is an abelian group
6-8 R is closed under •, contains a multiplicative identity (1 ≠ 0) and • is associative
9 multiplication distributes over addition
Subring
Let (R, +, •) be a ring. Let S ⊂ R and (S, +, •) be a ring - then we say S is a subring of R
Unit u ∈ R
Let R be a ring, then an element u is called a unit if there is some element v ∈ R such that uv = vu = 1
Unit group of R
Let R be a ring. The unit group of R is the set R* = {a ∈ R : a is a unit in R}
Field
A field (F, +, •) is a commutative ring such that every non-zero element is a unit
Euler’s φ-function
Let m ∈ ℤ. Euler’s φ-function φ(m) is the order of the unit group (ℤ / mℤ)*
Criterion for subring
A subset S of R is a subring iff it satisfies for a,b ∈ S a) 0,1 ∈ S b) a + b ∈ S c) -a ∈ S d) ab ∈ S Note ℤ, ℚ, ℝ and ℂ are all rings
SL₂(ℝ)
SL₂(ℝ) = {2x2 matrices with real entries and determinant = 1}
Quotient group (G/H, +)
Let (G,+) be an additive abelian group and H a subgroup. Then (G/H, +) is the set of congruence classes, ie G/H = {ā : a ∈ G}, and addition defined by (class of a) + (class of b) = (class of a+b)
Sym(A)
The set of bijections from A to itself - (Sym(A), ◦) is a group with idA as its identity element
n-th alternating group
An = {σ ∈ Sn : σ is even}
GL₂(ℝ)
GL₂(ℝ) = {2x2 matrices with real entries and non-zero determinant}
S_r group
S_r = {α ∈ ℂ : |α| = r}
SO₂(ℝ)
Set of matrices in form
(cos(θ) -sin(θ)
sin(θ) cos(θ))
where θ ∈ ℝ
