Algebraic Structure for Groups in Discrete Mathematics

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Algebraic structures are one of the fundamental concepts in abstract algebra. These structures are used to organize and understand the mathematical systems that operate under specific rules. One of the most common structures is groups. In this chapter, we will see the basics of algebraic structures on groups with examples for a better understanding.

What is an Algebraic Structure?

An algebraic structure is simply a set of elements which are combined with one or more operations that define how those elements interact. The set should be non-empty so that there should be at least one element in it. When we talk about "operations," we are referring to actions like addition, multiplication, or even more abstract things like set operations.

A binary operation is commonly used when defining algebraic structures. The Binary means that the operation is performed between two elements at a time. If we think about basic arithmetic: if we add two numbers, then it is a binary operation. For an algebraic structure to be valid. When we apply the operation on any two elements of the set, the result should still be an element from the same set (closure property).

Example of Closure in Natural Numbers

Let us take the set of natural numbers, where all numbers starting from 1 and going on forever (1, 2, 3, ...). If we take two natural numbers like 5 and 10, and we add them, we get 15. This result is also a natural number. So the natural numbers is "closed" under addition. But if we tried subtraction, like 5 minus 10, we get -5, which is not a natural number. So, natural number is not closed under subtraction.

What is a Group?

A group is a specific type of algebraic structure. It has a set and one binary operation, but it also satisfies a few more rules or properties that make it a group. These properties are −

• Closure − Like we said earlier, the result of applying the operation to any two elements from the set should stay in the set.
• Associativity − If you have three elements from the set, it does not matter how you group them when applying the operation. For example, if we are adding three numbers, it does not matter if we add the first two and then add the third, or if we add the last two first and then add that result to the first number.
• Identity Element − There should be one special element in the set that, when combined with any other element. It does not change the other element. For addition, this element is 0. For multiplication, it is 1.
• Inverse Element − For each element in the set, there should be another element in the set that, when combined with the first element using the operation, gives you the identity element. In addition, for example, every number has an inverse: the inverse of 5 is -5, because 5 + (-5) = 0, which is the identity for addition.

These four properties, closure, associativity, identity, and inverse they make a group.

Example of a Group Integers under Addition

Let us see some example to understand the idea. Take the set of all integers, which includes both positive and negative numbers (, -3, -2, -1, 0, 1, 2, 3, ). Now, consider the operation of addition.

• Closure − If we add any two integers, the result is always another integer. For example, -3 + 5 = 2, which is still an integer. So, integers are closed under addition.
• Associativity − Addition of integers is associative. Whether we calculate (-3 + 5) + 2 or -3 + (5 + 2), the result is still the same: 4.
• Identity Element − The identity element for addition is 0. No matter which integer we pick, adding 0 to it does not change the integer. For example, 5 + 0 = 5.
• Inverse Element − Every integer has an inverse. For instance, the inverse of 7 is -7, because 7 + (-7) = 0.

Since the set of integers with the operation of addition satisfies all these properties, it forms a group.

Example of Non-Groups

To understand them better let us see some non-groups. If we take natural numbers again, but this time under subtraction.

• Closure − As we saw earlier, natural numbers are not closed under subtraction. Subtracting two natural numbers not always give us a natural number. For example, 3 - 5 = -2, which is not a natural number.
• Inverse Element − For subtraction, the inverse of a number is its negative. But natural numbers do not include negative numbers, so there are no inverse elements in this set for subtraction.

Because these properties are missing, the set of natural numbers under subtraction does not form a group.

Types of Groups

In groups we see some special branches or types, depending on additional properties they may satisfy.

1. Abelian Groups

If the groups operation is commutative. So it does not matter in which order we apply the operation, the group is called an Abelian group. For example, the set of integers under addition is an Abelian group because 5 + 3 is the same as 3 + 5.

2. Non-Abelian Groups

In some groups, the order in which we apply the operation does matter. These are called non-Abelian groups. A classic example is the set of 2x2 matrices under matrix multiplication. The result of multiplying two matrices depends on their order, so this is a non-Abelian group.

More Examples of Groups

Let us see some other examples of groups.

Example 1: Real Numbers under Addition

The set of real numbers under the operation of addition is a group. It satisfies all four properties:

• Closure − Adding two real numbers gives a real number.
• Associativity − Real numbers follow the rule of associativity in addition.
• Identity Element − The number 0 is the identity element.
• Inverse Element − Every real number has an inverse, which is simply its negative.

Example 2: Real Numbers without Zero Under Multiplication

Let us see another example, here consider a set of real numbers, excluding 0, under the operation of multiplication, forms a group. Because:

• Closure − Multiplying two non-zero real numbers gives another non-zero real number.
• Associativity − Multiplication of real numbers is associative.
• Identity Element − The number 1 is the identity element for multiplication, because multiplying any number by 1 does not change it.

Inverse Element: Every non-zero real number has an inverse, which is its reciprocal. For example, the inverse of 5 is 1/5.

Conclusion

In this chapter, we explained the concept of algebraic structures, particularly focusing on groups. We understood the basic idea of an algebraic structure, which is a set with an operation that meets certain criteria. We also understood what makes a group, focusing on the four key properties, the closure, associativity, identity, and inverse.

By going through examples like integers under addition and real numbers under multiplication, we saw how these abstract concepts play out in real mathematical situations. Finally, we distinguished between types of groups, such as Abelian and non-Abelian groups.