13. Group Theory

So, let us start with the definition of group. So, what is a group? Imagine you are given a set $G$, which may or may not be finite, and you are given some binary operation. By binary operation, I mean it operates on 2 operands from $G$. So, $G$ along with the operation $\circ$ will be called a group if it satisfies certain properties, which we often call as group axioms.

A group is formed from a set combined with a binary operation. This operation takes two elements from the set and returns another element from the set. For the structure to be a group, certain conditions, known as group axioms, must be satisfied.

Think of a group like a team in a game where every team member must interact with each other according to specific rules. While there are many teams (sets), the way they play (binary operations) must follow certain predefined guidelines (group axioms).

The first axiom, which we call $A_1$, is the closure property, which demands that for any 2 operands $a, b$ from your set $G$, if you perform the operation $\circ$ on $a$ and $b$, the result should be an element of the set $G$ itself.

The closure property means that when you take any two elements from the group and apply the group operation, the outcome must also be an element of the same group. For example, if you add two numbers and the result is also a number within the same set, the closure property holds.

Imagine mixing colors. If you mix blue and yellow (two elements from your 'color group'), you get green, which is still a color. This illustrates the closure property—when combining elements, you still remain within the original set.

The second property or axiom is the associativity property, denoted by $A_2$, which demands that your operation $\circ$ should be associative, meaning the order of the operands does not matter. Namely, for every triplet of values $a, b, c$ from $G$, $(a \circ b) \circ c = a \circ (b \circ c)$.

Associativity stipulates that when performing the group operation on three elements, it does not matter how the operations are grouped. Thus, whether you first combine the first two elements and then the result with the third, or combine the last two first, the final result remains the same.

Think of assembling a sandwich. Whether you put the lettuce and tomato together first and then add the bread, or you add the bread first and then put in the lettuce and tomato, the sandwich will have the same contents either way.

The third property or axiom is the existence of an identity element, denoted by $A_3$, which requires that there should be a unique element $e$ present in $G$ such that for every group element $a$, performing the operation on $a$ and the identity element will give back $a$: $a \circ e = e \circ a = a$.

The identity element acts like a neutral player in the game of operations. It does not change any other element when used in the operation. For example, in addition, the number 0 is an identity element because when you add it to any number, the number remains unchanged.

Consider a remote control for a TV. The 'power on' button is like the identity element because when the TV is off, pressing it returns to the state where the TV is on—essentially the TV remains unchanged, representing the original state.

The fourth property, which is the existence of an inverse element, mandates that for every element $a$ in the set $G$, there should exist a unique element in $G$, denoted by $a^{-1}$, such that when performing the operation on $a$ and $a^{-1}$, you get the identity element: $a \circ a^{-1} = a^{-1} \circ a = e$.

For every element in the group, there exists another element that, when combined using the group operation, results in the identity element. This ensures that you can 'cancel out' an element to return to the identity of the group.

Think of reversing a step in a dance move. Each movement has a specific opposite that brings you back to your starting position, just like an inverse brings an element back to identity.

An important point to note here is that the axioms do not require the operation $\circ$ to be commutative. The group axioms only demand the operation $\circ$ be associative. This means the result of performing the group operation on $a$ and $b$ need not be the same as the result of performing the group operation on $b$ and $a$.

While many operations in mathematics are commutative (like addition), group operations do not require this. This means you can have a group where changing the order of elements changes the result.

Consider a recipe for baking; adding ingredients in a different order may yield a different outcome, like placing eggs before mixing flour might yield a thicker batter than the other way around.

So, if $G$ along with the binary operation $\circ$ satisfies all these 4 axioms, then ($G, \circ$) is a group. Even if one of these 4 properties is violated, the set $G$ along with operation $\circ$ would not constitute a group.

For a set and an operation to qualify as a group, all four axioms must be satisfied. If any axiom is broken, it cannot be classified as a group. This strict requirement ensures specific mathematical behaviors are preserved.

Imagine a sports team. If a player fails to follow any of the team's rules (axioms), the integrity of the team is compromised. Similarly, all rules must be followed for the group structure to hold.

So now, let us see some examples of groups. The set of integers $\mathbb{Z}$ under the operation + constitutes a group. Let's see whether all the 4 properties are satisfied or not...

In the following discussion, we illustrate how the set of integers under addition satisfies all four group axioms, proving its status as a group. We also explore additional sets and operations to see how they qualify or disqualify from being a group.

Using familiar concepts like adding numbers in a bank account illustrates group operations effectively; just like deposits and withdrawals must balance, the arithmetic properties correspond to our algebraic rules.

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