Abstract algebra

Introduction to groups Set and operation Consider a set A = {a,b}, and a binary operation ∗ defined on the set:

The rules / table define how two elements of the set combine under the operation.

Example: What nine rules are given by the table shown below?

We’re familiar with the set of real numbers (and its subsets, the rationals, the integers, and so on), and the operations of addition and multiplication. We know that various properties hold for the real numbers under these operations (commutative, associative, etc.). What we’re interested in with sets / operators in the abstract is whether or not they behave “nicely” like the reals - what properties do they satisfy?

Work through the properties defined below and write down some notes / examples!

• Closure: A set (A) is closed under a binary operation (∗) if for every x ∈ A, y ∈ A, we have x ∗ y ∈ A. Example: Does the set A = {a,b} with the operation ∗ defined by the table below satisfy the closure axiom?

• Associative property: The associative property (axiom) holds for a set (A) with a binary operation (∗) if for every x ∈ A, y ∈ A, z ∈ A we have x ∗ (y ∗ z)=(x ∗ y) ∗ z.

Example: Does the set A = {a,b} with the operation ∗ defined by the table below satisfy the associative axiom?

• Identity element: An identity element (denoted e) for a set (A) with a binary operation (∗) is an element such that for every x ∈ A, e ∗ x = x ∗ e = x. e iteself must be an element of A. Example: Does the set A = {a,b} with the operation ∗ defined by the table below have an identity element?

• Inverse: An element a ∈ A has an inverse a0 ∈ A under a binary operation (∗) if a ∗ a0 = a0 ∗ a = e, where e is the identity. If every x ∈ A has an inverse element, then A with ∗ satisfies the inverse property (axiom).

Example: Does each element in the set A = {a,b} have an inverse under ∗ ? Give the inverse of each element that has one. Group The four properties you just examined are the group properties.

A group < A, ∗ > is a set A with a binary operation ∗ defined on A that satisfies the four axioms below:

• A is closed under ∗.

• The operation ∗ is associative.

• There is an identity element (e) in A .

• Each x ∈ A has an inverse element x0 ∈ A.

Examples: (group or not a group?)

• < R, + >

• < Q \{0}, × >

• < Z, + >

• < Z, × >

• < A, ∗ > where ∗ is defined on A = {a,b} by

Example: Is < A, ∗ > a group, where A = {a,b,c} and ∗ is defined by the table? Which properties hold, and which fail? • Commutative property: The commutative property (axiom) holds for a set (A) with a binary operation (∗) if for every x ∈ A, y ∈ A we have x ∗ y = y ∗ x.

Example: Does the set A = {a,b} with the operation ∗ defined by the table below satisfy the commutative axiom?

• A group is an Abelian group if it satisfies the commutative axiom (as well as the four essential group properties).