Groups

In the field of mathematics known as algebra, one of the fundamental structures studied is called a "group." A group is a set equipped with an operation that combines any two elements to form a third element while satisfying specific properties known as group axioms. These properties provide a framework for understanding a wide variety of mathematical systems.

Definition of group

A group is defined on a set G with a binary operation * (often called multiplication) that combines any two elements a and b to form another element c = a * b such that the following properties, called axioms, are satisfied:

1. Closure: for all a, b in G, the result of the operation a * b is also in G.
2. Associativity: for all a, b and c in G,
   (a * b) * c = a * (b * c)
3. Identity element: There exists an element e in G such that for every element a in G,
   e * a = a * e = a
4. Inverse element: for every a in G, there exists an element b in G such that
   a * b = b * a = e where e is the identity element.

Examples of groups

Example 1: Integers under addition

Consider the set of integers Z under the operation of addition +. It forms a group because:

• Completion: Adding any two integers gives another integer.
• Associativity: For any integers a, b, and c, (a + b) + c = a + (b + c)
• Identity element: The identity element is 0 because a + 0 = a for any integer a.
• Inverse Element: The inverse of an integer a is -a because a + (-a) = 0.

Example 2: Symmetric group

The symmetric group on a set of elements is the group formed by all possible permutations of the elements. For a set of n elements, the symmetric group is often denoted as S n. This group is important because it describes how the elements can be rearranged.

The symmetric group on three elements, S 3

Consider the set {1, 2, 3}. The symmetric group S 3 has the following elements:

    {(), (1 2), (1 3), (2 3), (1 2 3), (1 3 2)}

Here, each element represents a permutation of the set. The identity permutation is represented by (), which means no change.

Properties of groups

Groups can display a variety of properties that make them interesting to study:

Abelian group

If the operation is commutative then the group is called abelian (or commutative). This means that for all a, b in G, a * b = b * a. The integers under addition are an example of an abelian group.

Non-abelian groups

When a group does not satisfy the commutative property, it is called non-abelian. The symmetric group S 3 is an example of a non-abelian group.

Finite and infinite groups

Groups can be classified as finite or infinite depending on the number of elements they contain:

• Finite Group: This group has a limited number of elements. For example, S 3 has 6 elements.
• Infinite group: This group has an infinite number of elements. The integers under addition form an infinite group.

Visualizing group structure

Groups can be visualized in a variety of ways, often depending on the context or the particular branch of mathematics being studied. One such method is called the Cayley table which is a grid that shows how the combination of elements in a group works for a finite group.

Kelly table example

Consider a small group with elements {e, a, b} and an operation *. The Cayley table of the group might look like this:

    

        

            
E
A
B
        
        

            
E
E
A
B
        
        

            
A
A
B
E
        
        

            
B
B
E
A
        
    

Here, each cell of the table represents the result of the operation for specific elements: a * b is found where the row labeled a meets the column labeled b.

Applications of groups

Groups are not just theoretical constructs; they play important roles in various areas of mathematics and applied science:

Symmetry and geometry

In geometry, groups describe the symmetries of objects. The set of all symmetries forms a group called the symmetry group. For example, the symmetries of a square include rotations and reflections that can be modeled as a group.

Cryptography

Some groups, such as those based on elliptic curves, have properties that make them useful for creating secure cryptographic systems that protect digital communications.

Physics

Groups form the basis of the mathematical foundation of many physical theories. For example, symmetry groups known as Lie groups are used in the standard model of particle physics.

Exercise: Explore groups

To further improve your understanding of groups, try these exercises:

1. Prove that the set of all non-zero rational numbers with multiplication is a group.
2. Write the Cayley table for the symmetry group of an equilateral triangle.
3. Find the inverse of each element in the group of integers under the modulus of sum 5.
4. Determine whether the set of all 2x2 invertible matrices over real numbers is a group under matrix multiplication. If yes, is it an abelian group?

Conclusion

Understanding groups is essential to studying algebra and its application to many other fields. Groups embody the idea of symmetry and structure in a mathematical system. Through the exploration of properties such as identity, inverse, and closure, groups provide insight into complex mathematical concepts. As you study groups, you build a foundation for analyzing more sophisticated algebraic systems and solving practical problems in science and engineering.

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