Abelian Group in Discrete Mathematics

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In discrete mathematics, "groups" are one of the core topics that help in understanding how different sets of elements can work under specific operations. Among these, the Abelian group, also known as commutative group, are special because of a special property it has: commutativity.

In this chapter, we will understand the basics of what an Abelian group is and how it’s different from other groups.

What is a Group?

For a basic recap on groups, we can say, a group is a set of elements that is combined with a binary operation (like addition or multiplication), but it also has to follow four key properties: Closure, Associativity, Identity Element, and Inverse Element. If a set with a binary operation satisfies these four properties, then it's a group.

What is an Abelian Group?

An Abelian group is just a special type of group that also satisfies the commutative property. It means the order in which we apply the operation does not matter.

Formally, we can state that for a group to be Abelian, it should satisfy the following −

For every pair of elements, A and B, in the group, applying the operation in both orders gives the same result. In simple terms, A + B should be the same as B + A.

If a group satisfies all four group properties and the commutative property, we can regard it as an Abelian group.

Note − The name "Abelian" comes from a famous mathematician named Niels Henrik Abel, who worked extensively in Group Theory.

Examples of Abelian Groups

Let us see some examples for Abelian Groups and how they work.

Example 1: Integers under Addition

Consider there is a set of all integers (…, -3, -2, -1, 0, 1, 2, 3, …) under the operation of addition. These are forming the Abelian group.

• Closure − If we add any two integers, we will always get another integer. For example, -3 + 5 = 2, which is still an integer.
• Associativity − Addition is associative, meaning (a + b) + c is the same as a + (b + c).
• Identity Element − The identity element for addition is 0 because adding 0 to any number does not change the number. For example, 5 + 0 = 5.
• Inverse Element − Every integer has an inverse under addition. For instance, the inverse of 7 is -7 because 7 + (-7) = 0.
• Commutativity − Addition of integers is commutative. It means, 5 + 7 is the same as 7 + 5.

Since the set of integers under addition satisfies all four group properties and is commutative, it is an Abelian group.

Example 2: Real Numbers under Multiplication

Let us see another example. Consider the set of all real numbers, excluding 0 (denoted as R*), under multiplication forms another Abelian group.

• Closure − Multiplying two real numbers gives another real number. For example, 2 * 5 = 10, and 1.5 * 3.2 = 4.8.
• Associativity − Multiplication is associative. (a * b) * c = a * (b * c).
• Identity Element − The identity element for multiplication is 1 because multiplying any number by 1 does not change the number.
• Inverse Element − Every non-zero real number has an inverse under multiplication. For example, the inverse of 5 is 1/5 because 5 * 1/5 = 1.
• Commutativity − Multiplication is commutative, meaning a * b is always the same as b * a.

Thus, the set of real numbers (except 0) under multiplication is an Abelian group.

Example 3: Non-Abelian Group of 2 × 2 Matrices

Let us see another example with matrix. Here the , let's look at an example where the commutative property fails: the set of 2 × 2 matrices under multiplication.

Matrix multiplication does not follow the commutative property. For example, if A and B are two 2 x 2 matrices, A * B is usually not the same as B * A. While matrices form a group under multiplication (since they satisfy closure, associativity, identity, and inverses). This lack of commutativity makes them a non-Abelian group.

Checking if a Group is Abelian

There are some way to check whether a group is Abelian or not. It is quite confusing that not all groups are Abelian. When solving problems in discrete math, It is important to always check whether a group is commutative before labelling it as Abelian. The best way to do this is to pick two elements from the group, apply the operation in both orders, and see if the results are the same.

Example of Integers under Subtraction

Let us check whether the set of integers under subtraction forms an Abelian group.

• Closure − Subtraction of two integers gives another integer. For example, 7 - 3 = 4, which is an integer.
• Associativity − Subtraction is not For example, (7 - 3) - 2 is 2, but 7 - (3 - 2) is 6.
• Identity Element − Subtraction does not have an identity element like 0 or 1.
• Inverse Element − Subtraction does not have an inverse like addition or multiplication.
• Commutativity − Subtraction is not 7 - 3 = 4, but 3 - 7 = -4, so the results are different.

Since it does not meet several of the group properties (and certainly not the commutative property), integers under subtraction do not form an Abelian group.

Importance of Abelian Groups in Mathematics

Abelian groups are quite useful concept in mathematics. They are used in number theory, cryptography, and the study of symmetry, etc. In fact, many of the groups we encounter in everyday mathematics are Abelian, which is why they are so commonly discussed in discrete math courses.

One reason Abelian groups are so important is that the commutative property often simplifies calculations. If we know that the order of operations does not matter, it can make solving problems much easier.

Conclusion

An Abelian group is a type of group that satisfies the commutative property. We first understood what a group is by looking at the properties it must satisfy: closure, associativity, identity, and inverses. Then, we understood what makes a group "Abelian", the commutative property.

In addition, we provided several examples, including integers under addition and real numbers under multiplication, to highlight how Abelian groups work in practice. Finally, we discussed how not all groups are Abelian by looking at examples like matrices and subtraction.