14. Cyclic Groups

Let us first prove that the identity element and inverse element are unique in any group. We first prove for the identity element. Let G be an abstract group. G has to have an identity element because that is one of the group axioms. We now have to prove that it has a unique identity element, i.e., G cannot have multiple identity elements.

In group theory, every group must have an identity element. An identity element is a special element that leaves other elements unchanged when combined with them using the group operation. To prove the uniqueness of the identity element within a group G, we assume there are two different elements, say e1 and e2, which both act as identities. By the properties of group operations, we can show that e1 must equal e2, leading to a contradiction. Thus, there must be exactly one identity element in the group.

Think of an identity element like a 'neutral button' for group activities, such as a switch that doesn’t change the state of a device when pressed. If you have two switches that claim to do nothing when pressed, yet they cannot both exist since pressing one would have to affect the outcome of the other.

Next, we show that every element in any abstract group G has a unique inverse element. You cannot have multiple inverse elements in the group. So again the proof will be by contradiction.

An inverse element for any element a in a group G is another element, say b, such that when a is combined with b using the group operation, it results in the identity element. To prove that each element has a unique inverse, we assume that there are two inverses, b1 and b2, for the same element a. By utilizing the inverse property, we conclude that b1 must equal b2, reaffirming that each element has one and only one inverse.

Imagine you have a lock that can only be opened with a single key. If two different keys are said to unlock the same lock, when you use one key and it works, the other key can’t work too, because a lock only has one mechanism confirming that a specific key opens it.

Now, we want to introduce a new operation in the group called group exponentiation. The group operation remains ∘, but we will be using that operation multiple times on an element of the group.

Group exponentiation is a way to express repeated application of the group operation on an element. For instance, if g is an element of a group, then g^n (read as g to the power n) refers to applying the group operation 'n' times on the element g. We formalize this recursively, where g^0 equals the identity element, and g^1 equals g itself, followed by g^n being defined as g multiplied by g^(n-1) for integers n.

Think of group exponentiation as a repetitive action, such as planting seeds. If you plant one seed (g^1), it will grow into a plant. Planting seeds multiple times (g^2, g^3, etc.) will lead to a garden that grows larger with each planting, where the first planting sets the stage for all subsequent growth.

Let G be a finite group. Now consider an arbitrary group element g. We define a function from the set of natural numbers to the group, mapping each number n to g^n.

The order of a group element g is defined as the smallest positive integer n such that g^n equals the identity element. If no such positive integer exists, we consider the order to be infinite. The concept of order is important because it helps to define the cyclic nature of groups derived from their elements through exponentiation.

Imagine a circular park where you can walk around. The order of a point on the path is like how many steps you need to take to return to the starting point. If you can step back to the start after a certain number of steps, that’s the order. If you can walk indefinitely without returning to the start, it’s like saying the order is infinite.

Now, let us define what we call a cyclic group. Let G be a group with some abstract operation ∘. The specialty of the group is that it has an element g which we call a generator.

A cyclic group is one that can be generated by a single element. This means that every element in the group can be expressed as the generator raised to various powers. If a group has a generator, it can be denoted as ⟨g⟩, indicating that g can produce all elements of the group through group exponentiation. Cyclic groups can be either finite or infinite.

Think of a musical scale as a cyclic group. If you can start at one note (the generator) and play different octaves (powers) of that note, you can produce all the notes in that scale. Each note you derive from the generator represents different elements of the group.

Consider the infinite group based on the set of integers with respect to addition. The claim is that the integer 1 constitutes your generator.

The set of all integers forms an infinite cyclic group where the number 1 generates the group. By repeatedly adding 1, or its negative counterpart, -1, we can generate any integer, thus covering the entire set. For instance, to generate the integer 5, we need to add 1 a total of five times (1+1+1+1+1).

Consider a circular track in an amusement park. If you have a friend who walks around the track once every minute (the generator), you can predict where your friend will be at any given minute based on multiples of walking once. If you only move forward or backward in time, you can eventually match any location on the track from just that one walk.

Assume G is a cyclic group of order n, and let g be a generator of G. My claim is that the order of the generator g is n.

In a finite cyclic group, the generator has the same order as the group itself. This is because if g generates the group, then by calculating all powers of g, you can recreate every element in the group without missing any. If there was a smaller order for g, there wouldn't be enough distinct elements created to cover the entire group.

Imagine a cooking recipe where the main ingredient might be flour. No matter how many different types of cakes you make (the group’s elements), flour (the generator) is essential. If flour wasn't in each recipe type, you wouldn’t be able to create cakes without it. Here, flour relates to the whole group, akin to the generator’s order covering the entire cyclic group.

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