Modern Algebra - Final Exam Review

Binary Operations (Associative, Communitative, & Closure)

set X is a function * : X x X → X where *(a,b) is denoted a * b

Associative: ∀ a,b,c ∈ X where a * (b * c) = (a * b) * c
Communiative: ∀ a,b ∈ X, a * b = b * a
Closure: Y ⊆ X, Y is closed under * if the map * to Y x Y is a binary operation on Y

Groups

A group is a set G together with a specified binary operation * on G:

(G1) * is associative

(G2) ∃ e ∈ G: ∀ a ∈ G, a*e = e * a = a (e is the identity)

(G3) ∀ a ∈ G, ∃ b ∈ G: a * b = e = b * a (b is an inverse of a)

Abelian Groups

A group with a communative operation

Finite Group

A group (G, *) for which the set G is a finite set

Group Facts

1. One identity element

2. ∀ a ∈ G, ∃ unique inverse of a (a^-1)

3. ∀ a ∈ G, (a^-1)^-1 = a

4. ∀ a ∈ G, (a * b)^-1 = b^-1 * a^-1

5. ∀ a₁, …, a_n ∈ G, the operation a₁*a₂ * … * a_n is independent of parentheses

6. ∀ a, b, c ∈ G, if a * b = a * c then b = c and if b * a = c * a, then b = c

Order of a group

the number of elements in G, |G| can be infinite or finite

Order of an element of a group

Cyclic Group

G = {aⁿ | n ∈ ℤ} for some a ∈ G. a is a cyclic generator of G written as G = <a>

* A finite grouo is cyclig iff ∃ a ∈ G: |a| = |G|