Groups

In all of our previous discussions, we work with a ring $R$ with two operations, addition and multiplication. In this chapter, we would like to discuss a structure that has only one binary operation, these are called groups.

Abelian Groups

A natural question to ask is: what happens if we remove the multiplication operation part of the ring structure? We can do this and still have a very interesting structure. We will call this structure an abelian group.

Definition

An abelian group is a set $G$ with a binary operation $+$ such that:

1. Closure: For all $a,b\in G$, $a+b\in G$.
2. Associativity: For all $a,b,c\in G$, $(a+b)+c=a+(b+c)$.
3. Commutativity: For all $a,b\in G$, $a+b=b+a$.
4. Identity: There exists an element $e\in G$ such that for all $a\in G$, $a+e=a$.
5. Inverse: For each $a\in G$, there exists an element $b\in G$ such that $a+b=e$.

One can compare it with the definition of a ring. The only difference is that we have removed the multiplication operation and the requirement for the distributive property. So all of our ring examples are also abelian groups. For example:

1. The set of integers $\mathbb{Z}$ with the operation of addition is an abelian group.
2. The set of rational numbers $\mathbb{Q}$ with the operation of addition is an abelian group.
3. The set of congruence classes $\mathbb{Z}/n\mathbb{Z}$ with the operation of addition is an abelian group.
4. The set of polynomials $F[x]$ with the operation of addition is an abelian group.
5. The congruence classes of polynomials $F[x]/(p(x))$ with the operation of addition is an abelian group.

The groups $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}$ has a special property, they are cyclic groups:

Definition

A cyclic group is an abelian group $G$ such that there exists an element $g\in G$ such that for all $a\in G$, there exists an integer $n$ such that $a=g^n$.

Not every abelian group is cyclic. For example, we have:

Proposition

The product ${\mathbb{Z}}_n\times {\mathbb{Z}}_m$ is cyclic if and only if $n$ and $m$ are coprime.

Groups

An abelian group is a special case of a more general structure called a group, we simply drop the requirement for commutativity:

Definition

A group is a set $G$ with a binary operation $\ast$ such that:

1. Closure: For all $a,b\in G$, $a\ast b\in G$.
2. Associativity: For all $a,b,c\in G$, $(a\ast b)\ast c=a\ast (b\ast c)$.
3. Identity: There exists an element $e\in G$ such that for all $a\in G$, $a\ast e=a$.
4. Inverse: For each $a\in G$, there exists an element $b\in G$ such that $a\ast b=e$.