§0 Introduction

Concept (Name)	All Numbers	Derivatives	Groups	Rings
What does this concept explain...? (Mathematics is a language for expressing ideas; what ideas do these words represent?)	Counting, quantity	Rate of change, linearization	Symmetry	Functions on spaces
Some mathematical results			The "algebraization" of geometry (from Descartes to the present), and the "geometrization" of algebra
Some applications (outside pure mathematics)			Noether's theorem (physics) RSA algorithm (cryptography) logic circuits as "cosheaves" homological shapes of data sets, etc.

§1 Groups

§1.1 Groups

Consider some object $X$.

What is a symmetry of $X$?

Usually, when we think about an object $X$, we already have some prior notion of its structure—its shape, distance, linear structure, and so on. For example, $X$ may be a circle, a triangle, a square, a metric space, or a vector space $V$, etc.

A symmetry of $X$ should be a map (why a map from $X$ to $X$? Because we are performing a transformation on $X$, while $X$ itself remains the same)

such that

1. it preserves the structure, for example $\operatorname{dist}(x, y)=\operatorname{dist}(\phi(x), \phi(y))$, or $\phi(x+y)=\phi(x)+\phi(y)$, and

2. the operation can be undone.

Now let us first look at this in the most general possible way, in order to gain some intuition.

Let $G={\phi}$ be the set of symmetries of $X$.

1. If $\phi_1,\phi_2$ preserve the structure, then their compositions $\phi_1\circ\phi_2$ and $\phi_2\circ\phi_1$ also preserve the structure. $\Rightarrow$ We can compose elements of $G$ $\Rightarrow$ $G\times G\xrightarrow{m}G$, with associativity;

2. "Doing nothing" should also be a symmetry of $X$. $\Rightarrow \mathrm{id}_X\in G$, and $\operatorname{id}_X\circ\phi=\phi\circ\mathrm{id}_X=\phi$;

3. Since $\phi$ can be undone, we should have $\phi^{-1}\in G$, so $\phi\circ\phi^{-1}=\mathrm{id}_X$, and $\phi^{-1}\circ\phi=\mathrm{id}_X$.

Seen from this most general point of view, does the definition of a group not now almost suggest itself?

§1.1.1 Definition of a Group

Definition 1.1 A group is an ordered pair

where $G$ is a set and $m$ is a map

such that

1. $m$ satisfies associativity, i.e.

i.e. $\left(g_1 \cdot g_2\right) \cdot g_3=g_1 \cdot\left(g_2 \cdot g_3\right)$ or $\left(g_1 g_2\right) g_3=g_1\left(g_2 g_3\right)$;

2. $\exists$ an element $1_G \in G$, called the identity, such that

i.e. $1_G \cdot g=g=g \cdot 1_G$ or $1_G g=g=g 1_G$;

3. $\forall g \in G$, $\exists$ an element $h\in G$ such that

i.e. $g\cdot h=1_G=h\cdot g$ or $gh=1_G=hg$. We usually write $g^{-1}:=h$ and call it the inverse of $g$.

§1.1.2 Examples of Groups

Example 1.1 Let

be the set of integers. Define

by

(i.e. integer addition).

For example,

Then $(G,m)$ is a group.

Proof.

$(\mathbb{Z},+)$ is a group because

(1) $m$ is associative:

$$(g+h)+k=g+(h+k).$$

(2) $0=1_G$ is the identity:

$$\begin{aligned} & m(0, g)=0+g=g \\ & m(g, 0)=g+0=g \end{aligned}$$

(3) Every element has an inverse:

$$m(g,-g)=g+(-g)=0.$$

$$~\tag*{$\square$}$$

Example 1.2 Let

be the set of integers, and let

For example,

Then $(G,m)$ is not a group.

Proof.

$(\mathbb{Z},\times)$ is not a group because not every element $z\in\mathbb{Z}$ has an inverse. For example, if $z=2$, its inverse would have to be $\dfrac{1}{2}$, which is not in $\mathbb{Z}$.

$$~\tag*{$\square$}$$

The two examples above show that

$(\mathbb{Z},+)$ is a group,

while

$(\mathbb{Z},\times)$ is not a group.

This illustrates that knowing $m$ is important. Nevertheless, we will often abbreviate our language and simply say "let $G$ be a group", omitting explicit mention of $m$.

Example 1.3 Let $G=\mathbb{R}\backslash{0}$ (the set of nonzero real numbers). Let

Then $G$ is a group. From now on, we denote it by $\mathbb{R}^{\times}$.

Proof.

$\mathbb{R}^{\times}$ is a group because

(1) multiplication of real numbers is associative;

(2) the number $1$ is the identity;

(3) for every $g\in\mathbb{R}\backslash{0}$, there exists $\dfrac{1}{g}$ such that $g\dfrac{1}{g}=\dfrac{1}{g}g=1$.

$$~\tag*{$\square$}$$

§1.1.3 Properties of Groups

Proposition 3.1 (Cancellation Law) Let $G$ be a group and let $g,h,k\in G$. Suppose

Then

Similarly,

Proof.

There exists $g^{-1}$ such that $g^{-1}g=1_G$.

$$\begin{aligned} g h=g k & \Rightarrow g^{-1}(g h)=g^{-1}(g k) \\ & \Rightarrow\left(g^{-1} g\right) h=\left(g^{-1} g\right) k \\ & \Rightarrow 1_G h=1_G k \\ & \Rightarrow h=k. \end{aligned}$$

We have used all the axioms of a group!

$$~\tag*{$\square$}$$

Remark: The cancellation law does not hold for arbitrary matrix multiplication unless the relevant matrices are invertible—for example, what happens if $g=0$?

Proposition 3.2 (Uniqueness of the Identity) The identity element of a group $G$ is unique. That is, if two elements $1_G$ and $1_G^{\prime}$ both satisfy the defining properties of the identity, then $1_G=1_G^{\prime}$.

Proof.

If $1_G$ is an identity, then for every $g\in G$,

$$1_Gg=g1_G=g.$$

In particular, setting $g=1_G^{\prime}$, we must have

$$1_G1_G^{\prime}=1_G^{\prime}.$$

On the other hand, if $1_G^{\prime}$ is also an identity, then

$$1_G1_G^{\prime}=1_G.$$

By transitivity,

$$1_G=1_G^{\prime}.$$

$$~\tag*{$\square$}$$

Proposition 3.3 (Uniqueness of Inverses) For every element $g\in G$, its inverse $g^{-1}$ is unique. That is, if elements $h$ and $h^{\prime}$ both satisfy the defining properties of the inverse of $g$, then $h=h^{\prime}$.

Proof.

Suppose $h$ and $h^{\prime}$ are both inverses of $g$. Then

$$gh^{\prime}=1_G.$$

Multiplying both sides on the left by $h$, we obtain

$$h\left(gh^{\prime}\right)=h.$$

By associativity, the left-hand side becomes

$$(hg)h^{\prime}=1_Gh^{\prime}=h^{\prime}.$$

Hence, by transitivity,

$$h^{\prime}=h.$$

$$~\tag*{$\square$}$$

§1.2 Abelian Groups

Example 1.4 Let $n\geqslant1$ and $n\in\mathbb{Z}$. Then

is a group, where

is given by matrix multiplication. This also illustrates that in general $gh\neq hg$.

Proof.

$GL_n(\mathbb{R})$ is a group because

(1) matrix multiplication is associative;

(2) the identity matrix is the identity element;

(3) $\operatorname{det}(g)\neq0\Rightarrow g$ is invertible.

$$~\tag*{$\square$}$$

Since matrix multiplication is not commutative, this shows that in general $gh\neq hg$. But what if the group multiplication is commutative? This leads to the following definition.

Definition 1.2 A group $G$ is called an Abelian group if, for all $g_1,g_2\in G$, we have $g_1g_2=g_2g_1$.

Definition 1.3 A group for which the group multiplication is not commutative for all elements is called a non-Abelian group.

§1.3 Cyclic Groups

Although Abelian groups already possess useful structure through commutativity, some Abelian groups are even simpler. What if the entire group could be generated by a single element? This leads to the notion of a cyclic group, one of the most fundamental examples of Abelian groups.

What makes cyclic groups special is that the entire group structure reduces to powers, or multiples, of a single element called a generator. This simplicity makes cyclic groups important building blocks in group theory and also useful tools in studying more complicated problems.

Definition 1.4 A group $G$ is called a cyclic group if and only if there exists an element $g\in G$, called a generator, such that every element of $G$ can be written as a power of $g$:

Example 1.5 (The Integers under Addition)

The group $(\mathbb{Z},+)$ is cyclic, generated by $1$:

Example 1.6 (A Group under Modular Arithmetic)

Under addition modulo $n$, the group $\mathbb{Z}/n\mathbb{Z}={0,1,\ldots,n}$ is cyclic. The element $1$ generates the whole group:

Proposition 1.4 Every cyclic group is Abelian.

Proof.

Let $G$ be a cyclic group generated by $g$. For any $x,y\in G$, there exist integers $m,n$ such that $g^m=x$ and $g^n=y$. Therefore,

$$xy=g^{m}g^{n}=g^{m+n}=g^{n}g^{m}=yx.$$

Hence $G$ is Abelian.

$$~\tag*{$\square$}$$

§1.4 Order of a Group

Definition 1.5 Let $G$ be a group. We define

to be the number of elements in $G$. We call $|G|$ the order of $G$.

Definition 2.3 For $g\in G$, consider the set

We define the order of $g$ to be

Example 1.7

• The order of $1_G\in G$ is $1$.

• Any $n\in\mathbb{Z}$, $n\neq0$, has infinite order.

• The matrix $\begin{pmatrix} 0 & 1\ 1 & 0 \end{pmatrix}\in GL_{2}(\mathbb{R})$ has order $2$, because

Thus ${\ldots,g^{-1},1,g,\ldots}={1,g}$.

Theorem 1.5 (Euler's Totient Function) The number of generators of a cyclic group of order $n$ is given by Euler's totient function $\varphi(n)$.

§1.5 The Center of a Group

To study non-Abelian groups, we often focus on subsets whose elements commute. One important concept is the center of a group.

Definition 1.7 The center of a group $G$, denoted by $Z(G)$, is defined by

Proposition 1.6 The center of an Abelian group is the whole group.

Proof.

The center of a group $G$ is defined by

$$Z(G) = \{z\in G \mid zg=gz~\text{for all}~g \in G\}.$$

In an Abelian group, every pair of elements commutes:

$$g_{1} \cdot g_{2}=g_{2}\cdot g_{1} \quad \text{for all}~g_{1}, g_{2} \in G.$$

Therefore every element of $G$ lies in the center, and hence

$$Z(G)=G.$$

$$~\tag*{$\square$}$$

The center of a group $G$ is the set of elements that commute with every element of $G$. This is not merely a collection of special elements: it itself forms an important structure inside $G$. We will introduce this structure in the next section.