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Recall from the Associativity and Commutativity of Binary Operations that an operation $\cdot$ on $S$ is said to be associative if for all $a, b, c \in S$ we have that $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ and $\cdot$ is said to be commutative if for all $a, b \in S$ we have that $a \cdot b = b \cdot a$.

On the Identity and Inverse Elements of Binary Operations page we said that an element $e \in S$ is the identity element of $S$ under $\cdot$ if for all $a \in S$ we have that $a \cdot e = a$ and $e \cdot a = a$.

We can now begin to describe our first type of algebraic structures known as groups, which are a set $G$ equipped with a binary operation $\cdot$ that is associative, contains an identity element, and contains inverse elements under $\cdot$ for each element in $G$.

Definition: A Group is a pair $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary operation on $G$ with the following properties:
1. For all $a, b, c \in G$, $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ (Associativity of $\cdot$).
2. There exists an $e \in G$ such that for all $a \in G$, $a \cdot e = a$ and $e \cdot a = a$ (The existence of an Identity Element).
3. For all $a \in G$ there exists an $a^{-1} \in G$ such that $a \cdot a^{-1} = e$ and $a^{-1} \cdot a = e$ (The existence of inverses).
Furthermore, if $G$ is a finite set then the group $(G, \cdot)$ is said to be a Finite Group and if $G$ is an infinite set then the group $(G, \cdot)$ is said to be an Infinite Group. More generally, the Order of $(G, \cdot)$ (or **Size of $(G, \cdot)$) is the size of $G$ and is denoted $| G |$.

When we use the multiplication symbol $\cdot$ to denote the operation on $G$, we often call $G$ a "multiplicative group". When the operation of the group is instead denoted by $+$ (instead of $\cdot$) then we often call $G$ an "additive group", and we write the inverse of each $a \in G$ as $-a$ (instead of $a^{-1}$).

Some of the sets and binary operations we have already seen can be considered groups. For example, $(\mathbb{R}, +)$ is a group under standard addition $+$ since the sum of any two real numbers is a real number, $+$, is associative, an additive identity $0 \in \mathbb{R}$ exists and inverse elements exist for every $a \in \mathbb{R}$ (namely $-a \in \mathbb{R}$).

Furthermore, $(\mathbb{Z}, +)$ is also a group under the operation of standard addition since the sum of any two integers is an integer, addition is associative, the additivity identity is $0 \in \mathbb{Z}$, and for all $a \in \mathbb{Z}$ we have $-a \in \mathbb{Z}$ as additive inverses.

We will examine many other (more interesting) groups later on, but for now, let's look at an example of a set and a binary operation that does NOT form a group.

Example 1

Consider the set of integers $\mathbb{Z}$ and define $*$ for all $a, b \in \mathbb{Z}$ by:

\begin{align} \quad a * b = a + 2b \end{align}

(Where the $+$ on the righthand side is usual addition of numbers). We will show that $(\mathbb{Z}, *)$ is NOT a group by showing that $*$ is not associative. Let $a, b, c \in \mathbb{Z}$. Then $*$ is not associative since:

\begin{align} \quad a * (b * c) = a * (b + 2c) = a + 2(b + 2c) = a + 2b + 4c \end{align}
\begin{align} \quad (a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c \end{align}

Clearly $a * (b * c) \neq (a * b) * c$ so $\mathbb{Z}$ does not form a group under the operation $*$.

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