📂Abstract Algebra

Abelian Groups in Abstract Algebra

Definition ¹

For a group $\left< G, \ast\ \right>$, if its two elements $a, b$ satisfy $a \ast\ b = b \ast\ a$, then $\left< G, \ast\ \right>$ is defined to be an Abelian group.

Explanation

Commutative can be understood as roughly meaning ’the commutative law holds’. In its English name, the word Abelian is used instead of Commutative, which is taken from the genius mathematician Abel. Of course, calling it an Abelian group in Korean causes no problem at all in conveying the meaning.

Once we reach the level of an Abelian group, it satisfies quite a lot of conditions, so it is not a structure that is hard to imagine. Let us look at an example that is a group but cannot be an Abelian group.

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

• The commutative law does not hold for matrix multiplication.

Usually, while encountering matrix operations, one would have treated it as quite important that the commutative law does not hold for multiplication. This means that, precisely because the commutative law is such a natural property for the numbers we deal with in everyday life, one should be careful about it. Conversely, this means that there are quite many examples that satisfy the commutative law, and those 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.

This is true even considering just the real numbers most familiar to us, and the same goes for complex numbers, rational numbers, integers, and so on. Usually it is a far more difficult task to find an example that is a group but not an Abelian group.