Definition Axioms and Basic Examples of Algebraic Groups
Let’s define a formal group! A group that is formed when people come together to accomplish specific goals and objectives is known as a formal group. An official group can be defined as a group that has particular structures and roles where the responsibilities of members of the group are defined.
What are Binary Operations?
A binary operation is just like an operation, except that it takes two elements, no more, no less, and combines the elements into one.
You already know a few binary operators, here they are :
5 + 3 equals 8.
4 × 3 equals 12.
4 - 4 equals 0.
These all take two numbers and then we combine them in different ways to get one number. Notice the last example, 4 - 4 equals 0. It still takes 2 elements, even if they are the exact same elements.
Here, we are going to discuss the formal definition of a group, introduction to groups.
Introduction to Groups
Now that we understand sets as well as operators, you know the basic building blocks that make up groups. Simply put:
We can define a group as a set combined with an operation. Let’s take, for example, the set of integers with addition.
We can't say much if we just know there are a set and an operator. What more could we describe? We need more information about the set as well as the operator. This is the reason why groups have restrictions placed on them. That is, they have more properties. Further, we are going through the formal definition of a group.
Formal Definition of a Group
Let’s know the formal definition of a group or the formal group meaning. A group is said to be a set G, combined with an operation *, such that:
The group contains an identity
The group contains inverses
The operation is associative
The group is closed under the operation.
That is the formal definition of a group. Let's look at those one at a time:
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1. The group contains an identity. If we use the operation on any element and the identity, we will get that same element back.
For the integers and addition, the identity is known to be "0". Because 5+0 = 5 and 0+5 = 5.
In other words, we can say that it leaves other elements unchanged when combined with them. There is only one identity element that is for every group.
The symbol for the identity element is e, or sometimes zero. But you need to start seeing the number zero as a symbol rather than a number. 0 is the symbol for denoting the identity, just in the same way e is. It's defined that way. In fact, many times mathematicians prefer to use the number 0 rather than e because it is much more natural.
Formal Statement: There exists an e in set G, such that a * e equals a and e * a equals a, for all elements a in G.
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2. The group contains inverses. Let’s suppose if we have an element of the group, there's another element of the group such that when we use the operator on both of them, we get e, the identity.
For the integers and addition, the inverse of the number 5 is -5. (because 5 + -5 equals 0)
In a similar manner for negative integers, the inverses are positives. -5 + 5 = 0, so the inverse of -5 is 5. In fact, if a is the inverse of suppose b, then it must be that b is the inverse of a. Inverses are unique. You can't name any other number x, such that 5 + x = 0 besides -5.
The notation that is used for inverses is a-1. So in the above example, a-1 = b. In a similar way, if we are talking about integers and addition, 5-1 equals -5.
Formal Statement: For all, a in G, there exists b in G, such that a * b equals e and b * a = e.
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3. Associative. You should have learned about associative rule in basic algebra. It basically explains that the order in which we do operations doesn't matter.
a * (c *d) equals (a * c) * d
Notice that we still went a...d...c. All that changed was the parentheses. We'll get back to this later.
4. Closed under the operation. Imagine you are closed inside a huge box. When you are on the inside of the box, you can't get to the outside. Similarly, if once you have two elements inside the group, no matter what the elements are, using the operation on them will not get you outside that group.
Suppose we have two elements in the group, a and b, it must be the case that a*b is also in the group. This is what we mean by closed. It is known as closed because from inside the group, we can't get outside of it.
And as with the earlier properties, the same is true with the integers as well as addition. If x and y are integers, x + y = z, it must be that z is an integer as well.
Formal Statement: For all elements let’s say, a, b in G, a*b is in G.
So, if you have a set as well as an operation, and you can satisfy every one of those conditions, then you have a Group.
FAQs on Introduction to Groups in Abstract Algebra
1. What is a group in mathematics?
A group is a set together with a binary operation that satisfies four properties: closure, associativity, identity, and inverses. In abstract algebra, a group is written as (G, *), where:
- Closure: For all a, b ∈ G, a * b ∈ G.
- Associativity: (a * b) * c = a * (b * c).
- Identity element: There exists e ∈ G such that a * e = e * a = a.
- Inverse element: For each a ∈ G, there exists b ∈ G such that a * b = b * a = e.
Groups are fundamental structures in algebra used to study symmetry, number systems, and transformations.
2. What are the four properties of a group?
The four properties of a group are closure, associativity, identity, and inverse. Specifically:
- Closure: The operation on any two elements of the set gives another element in the set.
- Associativity: The grouping of elements does not affect the result.
- Identity element: There exists an element e such that a * e = a for all a in G.
- Inverse element: Every element a has an inverse a⁻¹ such that a * a⁻¹ = e.
If any one of these properties fails, the structure is not a group.
3. What is an example of a group?
An example of a group is the set of integers under addition, written as (ℤ, +). It satisfies all group axioms:
- Closure: The sum of two integers is an integer.
- Associativity: (a + b) + c = a + (b + c).
- Identity: The identity element is 0, since a + 0 = a.
- Inverse: The inverse of a is −a, since a + (−a) = 0.
Thus, (ℤ, +) is a standard example used in introductory group theory.
4. What is the identity element in a group?
The identity element in a group is the element e such that a * e = e * a = a for every element a in the group. This element leaves every other element unchanged under the group operation.
- In (ℤ, +), the identity is 0.
- In (ℝ\{0}, ×), the identity is 1.
Every group has exactly one unique identity element.
5. What is the inverse of an element in a group?
The inverse of an element a in a group is an element a⁻¹ such that a * a⁻¹ = a⁻¹ * a = e, where e is the identity. Every element in a group must have an inverse.
- In (ℤ, +), the inverse of 5 is −5.
- In (ℝ\{0}, ×), the inverse of 3 is 1/3.
The inverse ensures that equations within the group can be solved.
6. What is the difference between a group and an abelian group?
The difference is that an abelian group satisfies the commutative property, while a general group may not. In an abelian group:
- Commutativity: a * b = b * a for all a, b in G.
For example, (ℤ, +) is abelian because addition is commutative, but the group of invertible matrices under multiplication is generally non-abelian.
7. How do you check if a set with an operation is a group?
To check if a set forms a group, verify closure, associativity, identity, and inverses. Follow these steps:
- Step 1: Test closure under the operation.
- Step 2: Prove associativity.
- Step 3: Find the identity element.
- Step 4: Show every element has an inverse.
If all four conditions hold, the structure is a group; if commutativity also holds, it is abelian.
8. What is the order of a group?
The order of a group is the number of elements in the group. It is denoted by |G|.
- If G has finitely many elements, it is a finite group.
- If G has infinitely many elements, it is an infinite group.
For example, the group ℤ under addition has infinite order, while the group {0,1,2} under addition modulo 3 has order 3.
9. What is a subgroup in group theory?
A subgroup is a subset of a group that is itself a group under the same operation. A subset H of G is a subgroup if:
- H is non-empty.
- H is closed under the operation.
- Every element of H has its inverse in H.
For example, the set of even integers forms a subgroup of (ℤ, +).
10. Why are groups important in mathematics?
Groups are important because they provide a unified way to study symmetry, algebraic structures, and transformations. Group theory is used in:
- Solving polynomial equations.
- Geometry and symmetry analysis.
- Number theory and modular arithmetic.
- Physics, especially quantum mechanics.
The concept of a group connects many areas of mathematics through a single abstract framework.
