📂Abstract Algebra

Commutative groups in Abstract Algebra

Definition ¹

A group $\left< G, \ast\ \right>$ is defined to be an Abelian group if for any two elements $a, b$ in $a \ast\ b = b \ast\ a$, $\left< G, \ast\ \right>$ satisfies the commutative property.

Explanation

The term “commutative” implies that the commutative law is applicable. In English, instead of Commutative, the term Abelian is used, named after the genius mathematician Abel. It is perfectly fine to refer to it as an Abel group in Korean for the sake of conveying the meaning.

By the time a group is an Abelian group, it has satisfied quite a complex structure, meaning it is not an unimaginable structure. Let’s look at an example where a group can be a group but not an Abelian group.

For a set of invertible square matrices $\text{GL}_{n} (\mathbb{R}) = \left\{ A \in \mathbb{R}^{n \times n} \ | \ \det A \ne 0 \right\}$, the group $\left< \text{GL}_{n} (\mathbb{R}) , \cdot \right>$ is not an Abelian group.

• The multiplication of matrices does not satisfy the commutative law.

The fact that the commutative law is not applicable in the multiplication of matrices is often taken as crucial when one first encounters operations with matrices. This emphasizes that the commutative law is a natural property in the numbers we deal with daily. Conversely, there are many examples where the commutative law is satisfied, and these examples are usually familiar to us.

The group $\left< \mathbb{R} , + \right>$ is an Abelian group.

• The addition of real numbers satisfies the commutative law.

Considering only the most familiar real numbers, the same also applies to complex numbers, rational numbers, and integers. It is often much more challenging to find an example of a group that is not an Abelian group.