Define Formal Group
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
1. What is a group in mathematics?
In abstract algebra, a group is a fundamental concept. It consists of a set of elements combined with a specific operation, known as a binary operation. For this set and operation to be called a group, they must together satisfy four essential properties: closure, associativity, the existence of an identity element, and the existence of an inverse element for every element in the set.
2. What are the four fundamental properties (axioms) that define a mathematical group?
For a set 'G' with a binary operation '*' to be considered a group, it must satisfy the following four axioms:
- Closure: For any two elements 'a' and 'b' in G, the result of the operation, a * b, is also in G.
- Associativity: For any elements 'a', 'b', and 'c' in G, the equation (a * b) * c = a * (b * c) holds true.
- Identity Element: There exists a unique element 'e' in G, called the identity element, such that for any element 'a' in G, a * e = e * a = a.
- Inverse Element: For each element 'a' in G, there exists a unique element 'b' in G, called the inverse of 'a', such that a * b = b * a = e, where 'e' is the identity element.
3. Can you provide some common examples of groups in mathematics?
Certainly. One of the simplest examples of a group is the set of all integers (ℤ) under the operation of addition. It satisfies all four group properties. Another key example is the set of non-zero rational numbers (ℚ*) under the operation of multiplication. Similarly, the set of rotations of a square that map the square onto itself forms a group under the operation of composition.
4. How does a binary operation relate to the definition of a group?
A binary operation is the cornerstone of a group's structure. It is a specific rule for combining any two elements from a set to produce a third element. Without a well-defined binary operation, the concept of a group cannot exist. The operation must be defined on the set and is what gets tested against the four group axioms (closure, associativity, identity, and inverse) to determine if the structure forms a group.
5. Why is the set of integers under the operation of multiplication NOT a group?
This is a classic example that helps clarify the group axioms. The set of integers (ℤ) under multiplication fails to be a group because it violates the inverse element property. While it has an identity element (1), most integers do not have a multiplicative inverse that is also an integer. For example, the inverse of the integer 2 is 1/2, which is not an integer. Since not every element has an inverse within the set, it is not a group.
6. What is the difference between an Abelian and a non-Abelian group?
The key difference lies in the commutative property. An Abelian group (or commutative group) is a group where the order of operation does not matter. For any two elements 'a' and 'b' in the group, a * b = b * a. The set of integers under addition is a common example. In a non-Abelian group, this property does not hold for at least one pair of elements. The group of 2x2 invertible matrices under matrix multiplication is a standard example of a non-Abelian group.
7. What is the importance of studying group theory in science and mathematics?
Group theory is crucial because it is the mathematical study of symmetry. Its importance extends far beyond pure mathematics. In physics, it helps describe the standard model of particle physics and the symmetries of physical laws. In chemistry, it is used to understand molecular symmetries and spectroscopic properties. In computer science, it is fundamental to cryptography and coding theory. Essentially, it provides a universal language for analysing symmetrical structures in any field.
