The document defines algebraic structures as collections of objects with operations that can be performed on them. It focuses on groups, rings, and fields. A group is defined as a set with an operation that satisfies four properties: closure, associativity, identity, and invertability. Examples of groups given include the integers, rational numbers, and real numbers under addition. It is noted that the integers also form an abelian group under addition. The document asks if the integers modulo 7 form a group and provides examples to verify group properties.

1. Algebraic
Structures

2. Algebraic Structures: Definition
 A collection of objects, and one or more operations that can be
performed on those objects. The objects can be numbers, but do
not have to be. It can be categorized algebraic structures based on
the properties of operations.
 There are many algebraic structures but let us focus only on
groups, rings, and fields.

3. Group: Definition
 A group is a set, G, together with an operation * (called the group
law of G) that combines any two elements a and b to form another
element, denoted 𝑎 ∗ 𝑏 𝑜𝑟 𝑎𝑏. To qualify as a group, the set and
operation (𝐺,∗), must satisfy four requirements known as group
axioms:
 CLOSURE
 ASSOCIATIVITY
 IDENTITY
 INVERTABILITY

4. Group Axiom 1: Closure
 For all 𝑎, 𝑏 𝑖𝑛 𝐺, the result of the operation 𝑎 ∗ 𝑏, is also in 𝐺.
For example:
 The integers under addition (ℤ, +) is closed.
1 and 2 are elements of ℤ.
1 + 2 = 3 is also an element of ℤ. And so, with all the positive and
negative integers.

5. Group Axiom 2: Associativity
 For all 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 𝑖𝑛 𝐺, (𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐).
 Example : In matrix multiplication (𝐴 ∗ 𝐵) ∗ 𝐶 = 𝐴 ∗ (𝐵 ∗ 𝐶 )
 (
1 2
2 3
*
1 2
0 3
) *
5 2
1 3
=
1 2
2 3
*(
1 2
0 3
*
5 2
1 3
)

6. Group Axiom 3: Identity
 There exists an element e in G such that, for every element a in G,
the equation 𝑒 ∗ 𝑎 = 𝑎 ∗ 𝑒 = 𝑎 holds. Such an element is unique and
thus one speaks about identity element.
 (The element e is called the identity element of G.)

7. Group Axiom 4: Invertability
 For all a in G, there exists an element b in G, commonly denoted
𝑎−1
(or −𝑎 , if the operation is denoted by “+”) such that
𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 = 𝑒, where 𝑒, is the identity element.
(The element b is called the inverse of a.)

8. Examples
Any set of elements with an operation that satisfies these
properties forms a group.
 ℤ is the set of all integers, with addition
 ℤn is the set of integers mod n, with addition
 ℚ the set of all rational numbers, with addition
 ℝ the set of nonzero real numbers, with multiplication
 GL (n, R), the set of n x n invertible matrices, with matrix multiplication

9. (ℤ, +) Z is the set of all integers, with addition.
Let us verify:
 CLOSURE : (-1)+5 = 4
 ASSOCIATIVITY : (9 + 4) + 5 = 9 + (4 + 5)
 IDENTITY : 9 + e = e + 9 = 9
 INVERTABILITY : 9 + b = e = 0

10. ℤn is the set of integers mod n, with addition
and multiplication

11. Is ℤ7
,∗ a group? Verify.

13. The integers under +
ℤ, +
The group ℤ satisfies an extra property not included in the definition
of group: Its operation is commutative. That is,
𝑎 + 𝑏 = 𝑏 + 𝑎 for all 𝑎, 𝑏 = ℤ.
A group whose operation is commutative is called abelian group, in
honor of the Norwegian mathematician Neils H. Abel (1802 – 1829).

15. The rational under +
ℚ, +
The system is denoted ℚ, + , or 𝑄 for short. The set of elements
ℚ = {𝑎
𝑏 |𝑎, 𝑏, ℰ ℤ, with b ≠ 0}.
The ℚ stands for Quotients; rational numbers are precisely those that
can be expressed as quotients of integers.
The operation is addition, defined by the rule
𝑎
𝑏
+
𝑐
𝑑
=
𝑎𝑑 + 𝑏𝑐
𝑏𝑑
Take this: ¾, 7/8

16. The real numbers under +
ℝ, +
Again, we take the group laws as well-known;
the identity is 0 and the inverse of an arbitrary real
number a is –a, as for integers. This group is
denoted ℝ, +

18. Non-examples of Groups:
But the natural numbers N with the operation + are not a
group

19. Task to do:
 Assignment: Answer Slide 11.
 Prepare for a quiz next meeting.