Abelian Group

An abelian group, also recognized as a commutative group, is an algebraic structure that not only meets the standard criteria of a group but also introduces an essential property: the commutativity of the group operation.

This signifies that for any pair of elements $a$ and $b$ within the group, swapping their order doesn't affect the outcome: $a \cdot b = b \cdot a$.

A structure must meet the following requirements to qualify as an abelian group:

Closure
The operation on any two elements $a, b$ in the group results in another element that is also part of the group.
Associativity
The operation is associative for any three elements $a, b, c$ in the group, meaning $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
Identity Element
There's an element in the group, usually denoted as $e$, such that for any element $a$ in the group, $a \cdot e = e \cdot a = a$.
Inverse Element
For every element $a$ in the group, there is an inverse $a^{-1}$ so that $a \cdot a^{-1} = a^{-1} \cdot a = e$, with $e$ being the identity element.

Alongside these foundational properties, what sets an abelian group apart is its adherence to commutativity.

In essence, for each pair of elements $a, b$ in the group, they obey the commutative property $a \cdot b = b \cdot a$.

Abelian groups are especially conducive to analysis and application across various disciplines due to these properties. Prominent examples include the set of all integers $\mathbb{Z}$ under addition and the set of non-zero rational numbers $\mathbb{Q}^*$ under multiplication.

Illustrating an Example

Consider the set of integers $\mathbb{Z}$ with addition serving as the operation to exemplify an abelian group.

This set $\mathbb{Z}$ encompasses all integers, positive, negative, and zero, represented as $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

Let's confirm that $\mathbb{Z}$ upholds the abelian group properties:

Closure
Any sum of two integers remains an integer.
For instance, $3 + (-5) = -2$, with $-2$ remaining within $\mathbb{Z}$.
Associativity
Integer addition is associative, as seen when $(2 + 3) + 1 = 2 + (3 + 1)$.
Identity Element
Zero serves as the additive identity, meaning adding zero to any integer yields that integer.
For example, $5 + 0 = 5$, applicable universally across integers.
Inverse Element
Each integer $a$ has an inverse $b = -a$ that sums to zero.
Thus, the inverse of $3$ is $-3$, since $3 + (-3) = 0$.
Commutativity
The sum of two integers is unaffected by their order, evidenced by $2 + 3 = 5$ and $3 + 2 = 5$.

These principles affirm that the group $ \mathbb{Z} , + $ with addition as its operation firmly stands as an abelian group.

Example 2

The set of integers $ \mathbb{Z} $ does not form a group when combined with multiplication $ \times $ because, outside of 1 and -1, integers do not possess multiplicative inverses.

The closest set to exhibit the properties of an abelian group is the finite set $\{1, -1\}$.

Let's explore how this set adheres to the abelian group properties under multiplication:

Closure
Multiplying any two elements within $\{1, -1\}$ always results in another element from the same set, demonstrating the group's multiplication table.

$ \times $	1	-1
1	1	-1
-1	-1	1

For instance: $$ 1 \times 1 = 1 $$ $$ 1 \times -1 = -1 $$ $$ -1 \times 1 = -1 $$ $$ -1 \times -1 = 1 $$

Associativity
The operation of multiplication is inherently associative.
For instance: $$ (1 \times -1) \times -1 = 1 $$ $$ 1 \times (-1 \times -1) = 1 $$
Identity Element
The identity element in multiplication is $1$, meaning for any element $a$ in the set, $a \times 1 = 1 \times a = a$, keeping $1$ and $-1$ unchanged when multiplied by $1$.
For instance: $$ 1 \times 1 = 1 $$ $$ -1 \times 1 = -1 $$
Inverse
Each element within the set has a multiplicative inverse also in the set. The inverse of $1$ is $1$ since $1 \times 1 = 1$, and the inverse of $-1$ is $-1$ since $-1 \times -1 = 1$.
For instance: $$ 1 \times 1 = 1 $$ $$ -1 \times -1 = 1 $$
Commutativity
The elements of the set engage in commutative multiplication.
For instance: $$ 1 \times -1 = -1 \times 1 = -1 $$

This illustrates that the set $\{1, -1\}$ under multiplication fulfills all criteria for being an abelian group.

However, it's crucial to understand that extending this group to include other integers under multiplication would not maintain the abelian group structure, as only $1$ and $-1$ have multiplicative inverses within the integer set.