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