Lecture 4

group

A set of objects, along with a binary operation

Closure, Associativity, identity element, inverse element

4 Group Properties

Field

A _______, denoted {F, +, ×}, is an integral domain whose elements satisfy the following additional property: multiplicative inverse

Commutative Ring

The set of all even integers, positive, negative, and zero, under the operations arithmetic addition and multiplication

NOT a field

The set of all even integers, positive, negative, and zero, under the operations arithmetic addition and multiplication is

ring

The set of all even integers, positive, negative, and zero, under the operations arithmetic addition and multiplication is a

integral domain

The set of all integers under the operations of arithmetic addition and multiplication

NOT a field

The set of all integers under the operations of arithmetic addition and multiplication is

set

objects

group

For a given value of N, the set of all N × N matrices over real numbers under the operation of matrix addition constitutes a

Subtraction

If the Group Operation is Referred to as Addition, then the Group Also Allows for

abelian group

If the operation on the set elements is commutative, the group is called an _______. An operation ◦ is commutative if a ◦ b = b ◦ a.

yes

Is the set of all integers, positive, negative, and zero, along with the operation of arithmetic addition an abelian group?

additional property 2

Let 0 denote the identity element for the addition operation. If a multiplication of any two elements a and b of R results in 0, that is if ab = 0 then either a or b must be 0

closed

R must be ______ with respect to the additional operator '×'

associativity

R must exhibit _______ with respect to the additional operator '×'.

closed, associativity, distribute

Ring Properties

multiplicative inverse

That is, if a ∈ F and a 6= 0, then there must exist an element b ∈ F such that ab = ba = 1

distribute

The additional operator (that is, the "multiplication operator") must _________ over the group addition operator

permutation group

The set Pn of permutations along with the composition-of-permutations operator is referred to as a

additional property 1

The set R must include an identity element for the multiplicative operation. That is, it should be possible to symbolically designate an element of the set R as '1' so that for every element a of the set we can say: a1 = 1a = a

group

The set of all 3 × 3 nonsingular matrices, along with the matrix multiplication as the operator forms a

field

The set of all complex numbers under the operations of complex arithmetic addition and multiplication is a

ring

The set of all integers under the operations of arithmetic addition and multiplication is a

Commutative Ring

The set of all integers under the operations of arithmetic addition and multiplication.

field

The set of all rational numbers under the operations of arithmetic addition and multiplication is a

field

The set of all real numbers under the operations of arithmetic addition and multiplication is a

ring

The set of all real numbers under the operations of arithmetic addition and multiplication is a

Commutative Ring

The set of all real numbers under the operations of arithmetic addition and multiplication.

integral domain

The set of all real numbers under the operations of arithmetic addition and multiplication.

0

When a group is denoted {G, +}, it is common to use the symbol ____ for denoting the group identity element

field

a _____ has a multiplicative inverse for every element except the element that serves as the identity element for the group operator

integral domain {R, +, ×}

commutative ring that obeys the following two additional properties

group

denoted by {G, ◦} where G is the set of objects and ◦ the operator

ordinary computing

division particularly is error prone and what you see is a high-precision approximation to the true result

finite field

finite set of numbers in which you can carry out the operations of addition, subtraction, multiplication, and division without error

Inverse element

for every a in the set, the set must also contain an element b such that a ◦ b = i assuming that i is the identity element.

Closure

if a and b are in the set, then the element a ◦ b = c is also in the set. The symbol ◦ denotes the operator for the desired operation

Identity Element

if for every a in the set, we have a ◦ i = a.

Commutative Rings

if the multiplication operation is commutative for all elements in the ring

Infinite Groups

meaning groups based on sets of infinite size, are rather easy to imagine

Associativity

means that (a ◦ b) ◦ c = a ◦ (b ◦ c).

infinite groups

set of all even integers — positive, negative, and zero — under the operation of arithmetic addition is a group.

infinite group

set of all integers — positive, negative, and zero — along with the operation of arithmetic addition constitutes a group

1. set 2. group 3. abelian group 4. ring 5. commutative ring 6. integral domain 7. field

stepping stones to understanding the concept of a finite field

composition of permutations

the composition π ◦ ρ means that we want to re-permute the elements of ρ according to the elements of π.

finite group

the set Pn of all permutations of the starting sequence sn can only be finite. As a result, Pn along with the operation of composition as denoted by '◦' forms a

ring

typically denoted {R, +, ×} where R denotes the set of objects, '+' the operator with respect to which R is an abelian group, the '×' the additional operator needed for R to form a

identity element

ρ ◦ π = π ◦ ρ

additive inverse

ρ1 + (−ρ1) = 0