14.7. Examples of Cyclic Groups

Let $G$ be a group with some abstract operation $\circ$. It may or may not be a finite group. The specialty of the group is that it has an element $g$ which we call a generator. It is called a generator because when you take different powers of this generator, again by power I mean group exponentiation, you will get all the elements of your group. That means, this element $g$ has the capacity to generate all the elements of your group by performing the group exponentiation on this generator. A group that has a generator $g$ is called cyclic and is represented by the notation $G = \langle g \rangle$.

A cyclic group is a special kind of group where all the elements can be generated from a single element called the generator. This means that if you start with the generator and apply the group operation repeatedly (which is a way of using the generator to 'create' new elements), you can create every element in the group. The notation $G = \langle g \rangle$ simply means the group $G$ is generated by the element $g$. This property helps in understanding the structure of the group, as it simplifies the analysis by focusing on just one element.

Think of a cyclic group like a revolving door: when you push it, it rotates in a circle around a fixed point. The door is the generator, and every time you push it (apply the group operation), you can generate all the positions (elements in the group) around the circle. No matter where you start pushing, you will eventually get back to your original position after a few pushes, just like how each element in the group can be reached from the generator.

Consider the infinite group, namely the group based on the set of integers with respect to the plus operation. My claim is that the integer 1 constitutes your generator. This is because if you take different powers of this element 1, it will give you all the elements of your set of integers. For instance, if you want to generate any arbitrary integer $n$ by computing some power of this element, it will be some $n \cdot 1$. So any integer can be reached by adding 1 repeatedly or even subtracting 1 repeatedly, showcasing the properties of a cyclic group.

The integers under addition form an infinite cyclic group with the integer 1 as a generator. This means that starting from 0 (the identity element), if you add 1 repeatedly (1 + 1 + 1 + ...), you can generate all the positive integers. Similarly, you can go in the opposite direction (subtracting 1) to generate all negative integers. Therefore, every integer can be seen as a multiple of 1, confirming the infinite nature of the group.

Imagine climbing a staircase. If each step up represents adding 1, you can reach any step (positive integers) just by moving up the staircase. Conversely, if you step down (subtracting 1), you can reach any position below the ground (negative integers). The integer 1 is like the step height: as long as you keep stepping up or down, you can visit every integer position either above or below the base (zero).

Consider the set of all integers modulo $p$ (a prime number). For example, if $p = 5$, then the set is {0, 1, 2, 3, 4} and if my operation is addition modulo 5, then this group is a cyclic group and actually has multiple generators; all the elements except the identity element 0 will be a generator for this group. For instance, 1, 2, 3, and 4 can all generate the entire group depending on how they are combined.

In a finite cyclic group such as the integers modulo a prime $p$, you can generate all group elements by starting with any non-zero element. In our case with $p = 5$, you can take 1 and generate 0 through 1 (0 times 1 = 0), 1 itself through 1 (1 time 1 = 1), and so on until you hit all the numbers up to $4$. Similarly, using 2 you can generate all the numbers as well, showing that there can be multiple generators in such groups.

Think of a clock with 5 hours showing the times 0, 1, 2, 3, and 4. If you start at 0, moving 1 hour ahead takes you to 1, then to 2, 3, and finally 4 before returning to 0, which is like adding 1 in mod 5 arithmetic. Alternatively, if you start at 2 and keep adding 2 (2 -> 4 -> 1 -> 3), you can still reach all the positions on the clock, which shows the property of different generators within the same cyclic group.

Assume $G$ is a cyclic group with $g$ as the generator and the order of the group is $n$. Then, the order of the generator is also $n$. This means that upon raising the generator to its order (n times), you will return to the identity element. The important conclusion drawn from this is that the generator will produce exactly $n$ distinct elements corresponding to the cyclic group's order.

The order of a cyclic group indicates how many distinct elements can be generated by its generator. If we assume that raising the generator to its order yields the identity element, it implies all other powers of the generator, from 0 to (n-1), will yield unique elements. This property ensures that cyclic groups are not just cyclical in nature but also maintain a structured relationship among their elements.

Consider a complete set of bicycle gears on a round gear mechanism. Each gear position represents a group element, and cycling through them is like applying powers to the generator. Once you complete a full rotation (having cycled through all the gears), you return to the starting position (identity element). Therefore, cycling through them gives you every positional change possible before coming back to where you started, analogous to how generators in cyclic groups work.

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