1. Groups

Groups are algebraic objects, and it is the study of these objects that forms a central part of abstract algebra, Group Theory. Before we get into our definition of a group, I will first introduce a few definitions that we will use to ultimately define a group.

Before we begin, you will need to remember that the symbol; $\forall$ means “for all elements”, $\exists$ means “there exists an element”, $\in$ means “in the set” and $\mathbb{Z}$ denotes the set of integers (whole numbers).

Definition 1.1: Sets, Elements, Non-Empty and the Empty Set

A Set is a collection of objects. The objects that make up a set are called the Elements of the set. A set is Non-Empty if it has one or more elements. The Empty Set denoted $\emptyset$ is the unique set with no elements.

Example 1.1: $X=\{1,2,3,4,5\}$ is a set and 1, 2, 3, 4 and 5 are its elements.

Definition 1.2: Cardinality

The Cardinality or size of a finite set A is the number of elements in the set A, denoted |a|. (we will look at the cardinality of infinite sets such as the Integers in future posts).

Example 1.2: For $X=\{1,2,3,4,5\}$ we have |X|=5, i.e. the set X has 5 elements.

Definition 1.3: Subset

Let A and B be sets. A is a Subset of B if every element of A is an element of B. This is denoted by A $\subseteq$ B.

Example 1.3: The set $Y=\{2,4\}$ is a subset of $X=\{1,2,3,4,5\}$ .

Definition 1.4: Composition Law

A Composition law (or Binary Function) on a set A is a rule $(\star)$ assigning to each $x\in A$ and $y\in A$ an element $z\in A$ such that $z=x\star y$ . We say that the set is closed

Example 1.4: Addition, +, on the set of all Integers, $\mathbb{Z}$ , is a compositon law. 1+2=3, we have assigned z=3 to x=1 and y=2.

Definition 1.5: Associative

A composition law $(\star)$ on a set A is associative if $(a \star b) \star c = a \star (b \star c), \forall a,b,c\in A$

Example 1.5: Addition on the set of all integers is Associative. e.g. (1+2)+3=3+3=6=1+5=1+(2+3)

We can write this more generally as follows: $\forall a,b,c \in \mathbb{Z}$ we have $(a+b)+c=a+(b+c)$ .

Definition 1.6: Commutative

A composition law $(\star)$ on a set A is commutative if $x\star y=y \star x, \forall x,y\in A$ .

Example 1.6: Addition on the Integers is Commutative. e.g 1+2=3=2+1.

More generally; $\forall a,b \in \mathbb{Z}$ we have $a+b=b+a$ .

Definition 1.7: Group

A pair $(G,\star)$ , where G is a non-empty set and $(\star)$ is a composition law, is a group if:

( $\star$ ) is associative: $a\star (b\star c)=(a\star b)\star c, \forall a,b,c \in \mathbb{Z}$ .
( $\star$ ) admits a neutral element: $\exists e_{G}\in G$ such that $e_{G}\star a=a\star e_{G}=a, \forall a\in G$
Every element of G has an inverse element in G: $\forall a\in G, \exists a^{-1}$ such that $a\star a^{-1}=a^{-1}\star a=e_{G}$

Example 1.7: The Integers, $\mathbb{Z}$ , endowed with addition, +, admits a neutral element, namely 0. e.g. 1+0=0+1=1.

More generally; $\forall a\in \mathbb{Z}, a+0=0+a=a$ .

Example 1.8: Every element of the Integers has an inverse element in the Integers with respect to the composition law of addition. e.g. 4+(-4)=(-4)+4=0.

More generally; $\forall a\in \mathbb{Z}, \exists (-a)$ such that $a+(-a)=0$ .

Now if we take a step back and look at examples 1.4, 1.5, 1.6, 1.7 and 1.8 all together we can see that the Integers endowed with Addition forms a group, which we denote by $(\mathbb{Z},+)$ .