The group theory is one of the most fundamental parts of our Number theory course. There is a wide use of groups in different areas of mathematics and Number theory is definitely not an exception. We will focus mainly on cyclic groups and their properties. This article should serve as a summary of some basic definitions, properties, and examples.

## Sources:

- Drapal, parts 1.1-1.
- Wiki post on group theory – you can find links to anything you should know ther
- Socratica has several videos dealing with group theory, especially this one dealing with cyclic groups, or this one explaining Lagrange’s theorem
- Numberphile video dealing with the Monster group

## Theory:

The notion of a **group** can be introduced in many ways. You can think of it as of a set with an associative binary operation , an inverse operation (for each there exists such that ), and one particular element of our set, such that is an identity element for , i.e., . We denote this situation by . or are among the most typical representatives of groups.

Let be a group. A non-empty subset is a **subgroup**, if is a group. Note that no mater how complicated group you have, some subgroups, namely and , are always present. These subgroups are called **trivial** for obvious reasons. Subgroups that are not trivial are called **proper** subgroups.

A group is called **cyclic**, if there exists an element such that . Such is called a **generator** of . is an example of an infinite cyclic group, because it is generated by the element , or . Another very important example is of the type , where all the operations proceed modulo .

Let be a group. The **order** **of** **a group** (denoted by ) is the cardinality of the set . For example, and the order of is, of course, infinite. The **order of an element** is the smallest such that . If we cannot find any such , then is said to be of an **infinite order**.

**LAGRANGE’S THEOREM**: Let be a subgroup of a finite group . Then divides .

Let and be two groups. A map is called a group **homomorphism**, if for any . Injective homomorphism is called a **monomorphism**, surjective homomorphism is an **epimorphism**, and if is a bijection, then we say that it is an **isomorphism**.

## Properties:

- Every subgroup of is equal to for some . Such is uniquely determined.
- Let and . Then is a subgroup if and only if , where .
- Every subgroup of a cyclic group is cyclic.
- Let be a cyclic group. If it is infinite, then . If is finite of order , then it isomorphic to . Moreover, in this case, contains a subgroup of order . If , then contains exactly one subgroup of order .

## Examples:

### Example 1:

Let . For each element , determine the order of . Deduce, whether is cyclic or not.

### Solution:

consists of elements and . The identity element is always of order . Secondly, and in , so is of order , and, hence, a generator of , i.e., is indeed a cyclic group. Next, , which implies that the order of is . Finally, , and . Therefore, is of order , which means that it is another generator.

### Example 2:

Consider . What are the subgroups of ?

### Solution:

We will use Lagrange’s theorem and the last of the Properties. is cyclic of order , so, the order of any subgroup must divide . Thus, potential orders of subgroups are and . However, we know that for any divisor , there exists a unique subgroup of order . There are two trivial subgroups, namely of order , and itself of order . Next, by the second property, two remaining subgroups are of order , and of order .

## Problems:

### Problem 1:

Decide whether , or are cyclic groups or not. If so, find all generators.

### Problem 2:

Use Lagrange’s theorem to prove that the order of any element in a group is a divisor of .