Intro to Abstract Algebra //

Question 1

Binary operation

Answer 1

A binary operation on a set S is a rule which for every two elements of S gives another element of S

Question 2

Commutative

Answer 2

A binary operation ◦ is commutative on S if a ◦ b = b ◦ a for all a, b ∈ S

Question 3

Associative

Answer 3

A binary operation ◦ is associative on S if (a ◦ b) ◦ c = a ◦ (b ◦ c) for all a, b, c ∈ S

Question 4

9 axioms of the real numbers

Answer 4

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

Question 5

Composition

Answer 5

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))

Question 6

Group (G, ◦)

Answer 6

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.

Question 7

Abelian group

Answer 7

A group (G, ◦) is abelian if, in addition to the first four properties, it is also commutative

Question 8

Order of an element

Answer 8

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

Question 9

Order of a group

Answer 9

The order of a group G is the number of elements that G has, denoted by |G| or #G

Question 10

Subgroup H of G

Answer 10

Let (G, ◦) be a group. Let H be a subset G, then if (H, ◦) is also a group, H is a subgroup of G

Question 11

Lagrange’s theorem v1

Answer 11

Let G be a finite group, and let g be an element of G. Then the order of g divides the order of G

Question 12

Lagrange’s theorem v2

Answer 12

Let G be a finite group, and H a subgroup of G. Then the order of H divides the order of G

Question 13

Cyclic subgroup ⟨g⟩

Answer 13

The subgroup ⟨g⟩ = {g^n : g ∈ ℤ} is called the cyclic subgroup generated by g

Question 14

Isomorphism

Answer 14

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)

Question 15

Left coset of H in G

Answer 15

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

Question 16

Right coset of H in G

Answer 16

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

Question 17

Index of H in G

Answer 17

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]

Question 18

Lagrange’s theorem v3

Answer 18

Let G be a finite group and H a subgroup. Then |G| = [G : H] * |H|

Question 19

Congruent modulo H

Answer 19

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)

Question 20

Congruence class

Answer 20

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)}

Question 21

Injective

Answer 21

f is injective if whenever a1, a2 ∈ A, a1 ≠ a2 then f(a1) ≠ f(a2)

Question 22

Surjective

Answer 22

f: A → B is surjective if for every b ∈ B, there is some element a ∈ A such that f(a) = b

Question 23

Pigeonhole principle

Answer 23

Let A be a finite set and let f be a function from A to A. Then f is injective ⇔ f is surjective

Question 24

Identify map on A

Answer 24

The identity map on a set A is the map idA : A → A satisfying idA(x) = x for all x ∈ A

Question 25

Invertible

Answer 25

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

Question 26

n-th symmetric group

Answer 26

The group Sym({1,2,...,n}), denoted Sn

Question 27

Ring (9 properties)

Answer 27

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

Question 28

Subring

Answer 28

Let (R, +, •) be a ring. Let S ⊂ R and (S, +, •) be a ring - then we say S is a subring of R

Question 29

Unit u ∈ R

Answer 29

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

Question 30

Unit group of R

Answer 30

Let R be a ring. The unit group of R is the set R* = {a ∈ R : a is a unit in R}

Question 31

Field

Answer 31

A field (F, +, •) is a commutative ring such that every non-zero element is a unit

Question 32

Euler's φ-function

Answer 32

Let m ∈ ℤ. Euler's φ-function φ(m) is the order of the unit group (ℤ / mℤ)*

Question 33

Criterion for subring

Answer 33

``` 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 ```

Question 34

SL₂(ℝ)

Answer 34

SL₂(ℝ) = {2x2 matrices with real entries and determinant = 1}

Question 35

Quotient group (G/H, +)

Answer 35

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)

Question 36

Sym(A)

Answer 36

The set of bijections from A to itself - (Sym(A), ◦) is a group with idA as its identity element

Question 37

n-th alternating group

Answer 37

An = {σ ∈ Sn : σ is even}

Question 38

GL₂(ℝ)

Answer 38

GL₂(ℝ) = {2x2 matrices with real entries and non-zero determinant}

Question 39

S_r group

Answer 39

S_r = {α ∈ ℂ : |α| = r}

Question 40

SO₂(ℝ)

Answer 40

Set of matrices in form (cos(θ) -sin(θ) sin(θ) cos(θ)) where θ ∈ ℝ