Group Theory 1: An Overview

Before we begin to discuss the main subject, I would like to give an overview of what we study in group theory or more generally in algebra.

Algebra (as a subject) is the study of algebraic structures. So, what is an algebraic structure? An algebraic structure or an algebra in short $\underline{A}$ is a non-empty set $A$ with a binary operation $f$. (In general, a set of operations but here I consider only one binary operation for simplicity.) $\underline{A}$ is usually written as the ordered pair
$$\underline{A}=(A,f).$$
A binary operation $f$ on a set $A$ is a function $f: A\times A\longrightarrow A$. An example of a binary operation is addition $+$ on the set of integers $\mathbb{Z}$. $+$ is a function $+:\mathbb{Z}\times\mathbb{Z}\longrightarrow\mathbb{Z}$ defined by $+(1,1)=2$, $+(1,2)=3$, and so on. We usually write $+(1,1)=2$ as $1+1=2$. In general, one may consider an $n$-ary operation $f:\prod_{i=1}^n A\longrightarrow A$, where $\prod_{i=1}^n A$ denotes the $n$-copies of $A$, $A\times A\times\cdots\times A$.

There are many different kinds of algebras. Let me mention some of algebras with a binary operation here. For starter, $(A,\cdot)$, a non-empty set $A$ with a binary operation $\cdot$ is called a groupoid. A groupoid $(A,\cdot)$ with associative law
$$(ab)c=a(bc)$$
for any $a,b,c\in A$ is callaed a semigroup. If the semigroup has an identity element $e\in A$ i.e.
$$ae=ea=a$$
for any $a\in A$, it is called a monoid. If for every element $a$ of the monoid $A$, there exists an inverse element $a^{-1}\in A$ such that $aa^{-1}=a^{-1}a=e$, the monoid is called a group. A group $(A,\cdot)$ with commutative law i.e.
$$ab=ba,$$
for any $a,b\in A$ is called an abelian group named after a Norwegian mathematician Niels Abel. Note the inverse ${}^{-1}$ can be regarded as an operation on $A$, a unary operation ${}^{-1}: A\longrightarrow A$ defined by ${}^{-1}(a)=a^{-1}$ for each $a\in A$. The identity element $e$ can be also regarded as an operation, a nullary operation $e:\{\varnothing\}\longrightarrow A$. Thus, formally a group can be written as $(A,\cdot,{}^{-1},e)$, a quadrupple of a nonempty set, a binary operation, a unary operation, and a nullary operation.

Now we know what a group is and apparently, group theory is the study of groups. But what exactly are we studying there? What I am about to say is not really limited to group theory but commonly applies to studying other algebraic structures as well. There are briefly two main objectives with studying groups. One is the classification of groups. This becomes particularly interesting with groups of finite order. Here the order of a group means the number of elements of a group. We would like to answer the question ``how many different groups of order $n$ are there for each $n$ and what are they?'' The classification gets harder as $n$ gets larger. There are groups with the same order that appear to be different. But don't be deceived by the appearance. They may actually be the same group. What do we mean by same here? We say two groups of the same order same if there is a one-to-one and onto map (a bijection) that preserves operations. Such a map is called an isomorphism. It turns out that if a map $\psi: G\longrightarrow G'$ from a group $G$ to another group $G'$ preserves binary operation, it automatically preserves unary and nullary operations. Here we mean preserving binary operation by
$$\psi(ab)=\psi(a)\psi(b)$$
for any $a,b\in G$. A map $\psi: G\longrightarrow G'$ which preserves binary operation is called a homomorphism. If a homomorphism $\psi: G\longrightarrow G'$ is one-to-one and onto, it is an isomorphism. An isomorphism $\psi: G\longrightarrow G$ from a group $G$ onto itself is called an automorphism. In group theory, if there is an isomorphism from a group to another group, we do not distinguish them no matter how different they appear to look. The other objective in studying group theory is to discover new groups from old groups. Some of the new groups may be smaller in size than the old ones. Here we mean smaller in size by having a smaller number of elements i.e. having a lesser order. Some examples are subgroups and quotient groups. Some of the new groups are larger in size than the old ones. An example is direct products. Subgroups, quotient groups (also called factor groups), direct products are the things we will study as means to get new groups from old groups.

Group theory has great significance in geometry. In geometry, symmetry plays an important role. There are different types of symmetries: reflections, rotations, and translations. An interesting connection between geometry and group theory is that these symmetries form a group. The most general such group of finite order is called a symmetric group. In mathematics, the embedding theorem is conceptually and philosophically important, although it may be practically less important. When we study mathematics, we often feel that the structures we study are highly abstract and we feel like they only exist in our consciousness but not in the physical world. The embedding theorem tells that those abstract structures we study are indeed substructures of a larger structure that we are familiar with in the physical world. The embedding theorem implicates that we are not making up those abstract mathematical structures but we are merely discovering them which already exist in the universe. This kind of view point is called Mathematical Platonism. It turns out that there is an embedding theorem in finite group theory, namely every group of finite order is a subgroup of a symmetric group. The embedding theorem is called Cayley theorem. This means that the study of finite groups boils down to studying symmetric groups.