📂Abstract Algebra

In group Theory in Abstract Algebra

Definition ¹

For elements $a$ and the identity element $e$ of a monoid $\left< G, \ast\ \right>$, if there exists $a '$ satisfying $a \ast\ a’ = a’ \ast\ a = e$, then $\left< G, \ast\ \right>$ is defined as a group. That is, a group is a binary operation structure that satisfies the following properties:

• (i): The operation is associative.
• (ii): An identity element exists for all elements.
• (iii): An inverse element exists for all elements.

Explanation

Starting from magma and moving through semigroup, monoid, we finally arrive at group. It may not seem like much, but compared to magma, the conditions have significantly expanded. It must be closed under the operation, associative, and have both an identity and inverse elements, so not just anything can be called a group.

The reason for studying groups is that they are much simpler and easier than other algebraic structures. If the conditions are less than that of a group, useful properties decrease, and if the conditions are more, the applications narrow.

Most of the algebraic structures of interest in algebra fundamentally rely on groups, and algebra is applied not only in pure mathematics like number theory but also in applied mathematics that meld into daily life, such as in cryptography. Surprisingly, group theory is also used in physics outside of mathematics.

Let’s look at an example where it becomes a monoid but not a group.

For the set of regular matrices $\mathbb{R}^{n \times n}$, the monoid $\left< \mathbb{R}^{n \times n} , \cdot \right>$ is not a group.

• For the matrix $A \in \mathbb{R}^{n \times n}$, if $\det A = 0$, then $A^{-1}$ does not exist.

Of course, with certain restrictions on the set, it can become a group.

For the set of regular matrices with inverses $\text{GL}_{n} (\mathbb{R}) = \left\{ A \in \mathbb{R}^{n \times n} \ | \ \det A \ne 0 \right\}$, the monoid $\left< \text{GL}_{n} (\mathbb{R}) , \cdot \right>$ is a group.

• The sub-monoid $\left< \text{GL}_{n} (\mathbb{R}) , \cdot \right>$ of $\left< \mathbb{R}^{n \times n} , \cdot \right>$ has an inverse for multiplication by definition $\text{GL}_{n} (\mathbb{R})$, therefore it is a group.

Symmetry?

When grasping the concept of a group, the discussion often starts with symmetry or is entirely constructed upon mathematical definitions.

As an example of symmetry, actions such as rotating or leaving a Rubik’s Cube as is (identity), or reversing (inverse), are often mentioned. However, these explanations make it easy to understand that structures with symmetry have the structure of a group but difficult to comprehend that the structure of a group possesses symmetry. It’s good to think of the concept of symmetry as relating to the concept of inverses in groups.

If $a$ exists, then by the definition of a group, the corresponding $a '$ must exist.

On the other hand, as $a '$ also has a corresponding $a’’=a$, it is natural to think of this relationship as symmetry. The difference between a monoid and a group is solely the inverse, so the concept and definition are more validly matched.

Now that we’re talking about symmetry, let’s look at a perfect example of a group with symmetry.

The monoid $\left< \mathbb{Z} , + \right>$ is a group.

• For integers $a$, $-a$ always has an inverse satisfying $a + (-a) = 0$.

If $1$ exists, then symmetrical $-1$ exists around the identity element $0$, for $-2$, $2$ exists, for… $n$, $-n$ exists. Thinking in terms of symmetry, this is quite a natural example.

Generally, when dealing with groups alone, group $\left< G, \ast\ \right>$ is simply represented as $G$, and operations are written as $\cdot$ unless otherwise mentioned. However, in contexts like $\left< \mathbb{Z} , + \right>$ where addition is clear, $+$ is used, adapting the appropriate operation as needed.