In this lecture, we study some basic number theory as it is needed to study group theory.

Let $\mathbb{Z}$ denote the set of integers. $\mathbb{Z}$ satisfies well-ordering principle, namely any non-empty set of nonnegative integers has a smallest member.

One of the most fundamental theorems regarding numbers is Euclid’s Algorithm. Although we will not discuss its proof, it can be proved using well-ordering principle.

Theorem. [Euclid’s Algorithm] If $m$ and $n$ are integers with $n>0$, then $\exists$ integers $q$ and $r$ with $0\leq r<n$ such that $m=qn+r$.

Euclid’s algorithm hints us how we can define the notion that one integer divides another.

Definition. Given $m\ne 0, n\in\mathbb{Z}$, we say $m$ divides $n$ and write $m|n$ if $n=cm$ for some $c\in\mathbb{Z}$.

Examples. $2|14$, $(-7)|14$, $4|(-16)$.

If $m|n$, we call $m$ a divisor or a factor of $n$, and $n$ a multiple of $m$. To indicate $m$ is not a divisor of $n$, we write $m\not|n$. For example, $3\not|5$.

Lemma. The following properties hold.

(a) $1|n$ $\forall n$.

(b) If $m\ne 0$ then $m|0$.

(c) If $m|n$ and $n|q$, then $m|q$.

(d) If $m|n$ and $m|q$ then $m|(\mu n+\nu q)$ $\forall \mu,\nu$.

(e) If $m|1$ then $m=\pm 1$.

(f) If $m|n$ and $n|m$ then $m=\pm n$.

Definition. Given $a,b$ (not both 0), their greatest common divisor (in short gcd) $c$ is defined by the following properties:

(a) $c>0$

(b) $c|a$ and $c|b$

(c) If $d|a$ and $d|b$ then $d|c$.

If $c$ is the gcd of $a$ and $b$, we write $c=(a,b)$.

$(24,9)=3$. Note that the gcd 3 can be written in terms of 24 and 9 as $3\cdot 9+1\cdot (-24)$ or $(-5)9+2\cdot 24$. In general, we have the following theorem holds.

Theorem. If $a,b$ are not both 0, their gcd exists uniquely. Moreover, $\exists m,n\in\mathbb{Z}$ s.t. $c=ma+nb$.

Now let us talk about how to find the gcd of two positive numbers $a$ and $b$. W.L.O.G. (Without Loss Of Generality), we may assume that $b<a$. Then by Euclid’s algorithm we have
$$a=bq+r,\ \mbox{where}\ 0\leq r<b.$$
Let $c=(a,b)$. Then $c|r$, so $c$ is a common divisor of $b$ and $r$. If $d$ is a common divisor of $b$ and $r$, it is also a common divisor of $a$ and $b$. This implies that $d\leq c$ and so $c=(b,r)$. Finding $(b,r)$ is of course easier because one of the numbers is smaller than before.

Example. [Finding GCD]
$$
\begin{aligned}
(100,28)&=(28,16)\ &(100&=28\cdot 3+16)\\
&=(16,12)\ &(28&=16\cdot 1+12)\\
&=(12,4)\ &(16&=12\cdot 1+14)\\
&=4.
\end{aligned}
$$
By working backward, we can also find integers $m$ and $n$ such that
$$4=m\cdot 100+n\cdot 28.$$
\begin{align*}
4&=16+12(-1)\\
&=16+(-1)[28+(-1)16]\\
&=(-1)28+2\cdot 16\\
&=(-1)28+2[100+(-3)28]\\
&=2\cdot 100+(-7)28.
\end{align*}
Therefore, $m=2$ and $n=-7$.

Definition. We say that $a$ and $b$ are relatively prime if $(a,b)=1$.

Theorem. The integers $a$ and $b$ are relatively prime if and only if $1=ma+nb$ for some $m$ and $n$.

Theorem. If $(a,b)=1$ and $a|bc$ then $a|c$.

Theorem. If $b$ and $c$ are both relatively prime to $a$, then $bc$ is also relatively prime to $a$.

Definition. A prime number, or shortly prime, is an integer $p>1$ such that $\forall a\in\mathbb{Z}$, either $p|a$ or $(p,a)=1$.

Suppose that $p$ is a prime as defined above and $p=ab$, where $1\leq a<p$. Then $p\not|a$ since $a<p$, so $(p,a)=1$. This implies that $p|b$. On the other hand, $b|p(=ab)$ and hence $p=b$ and $a=1$. So, the above definition coincides with the definition of a prime we are familiar with.

Theorem. If $p$ is a prime and $p|a_1a_2\cdots a_n$, then $p|a_i$ for some $i$ with $1\leq i\leq n$.

Proof. If $p|a_1$, we are done. If not, $(p,a_1)=1$ and so $p|a_2a_3\cdots a_n$. Continuing this, we see that $p|a_i$ for some $i$.

Regarding primes, we have the following theorems.

Theorem. If $n>1$, then either $n$ is a prime or the product of primes.

Theorem. [Unique Factorization Theorem] Given $n>1$, there is a unique way to write $n$ in the form $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_1<p_2<\cdots<p_k$ are primes and the exponents $a_1,\cdots,a_k$ are all positive.

Theorem. [Euclid] There is an infinite number of primes.

In my previous notes here, I mentioned some about logical symbols. The logical symbols I will use often are $\forall$ which means “for all”, “for any”, “for each”, or “for every” depending on the context, $\exists$ which means “there exists”, and $\ni$ which means “such that” (don’t be confused with $\in$ which means “be an element of”). We also use s.t. for “such that.” There are also $\Longrightarrow$ which means “implies” and $\Longleftrightarrow$ which means “if and only if.” I guess these pretty much cover what we use most of time.

Now lets review about functions in a more formal way. Let $X$ and $Y$ be two non-empty sets. The the Cartesian product $X\times Y$ of $X$ and $Y$ is defined as the set
$$X\times Y=\{(x,y): x\in X,\ y\in Y\}.$$
A subset $f$ of the Cartesian product $X\times Y$ (we write $f\subset X\times Y$) is called a graph from $X$ to $Y$. A graph $f\subset X\times Y$ is called a function from $X$ to $Y$ (we write $f: X\longrightarrow Y$) if whenever $(x,y_1),(x,y_2)\in f$, $y_1=y_2$. If $f: X\longrightarrow Y$ and $(x,y)\in f$, we also write $y=f(x)$. A function $f: X\longrightarrow Y$ is said to be one-to-one or injective if whenever $(x_1,y),(x_2,y)\in f$, $x_1=x_2$. This is equivalent to saying $f(x_1)=f(x_2)$ implies $x_1=x_2$. A function $f: X\longrightarrow Y$ is said to be onto or surjective if $\forall y\in Y$ $\exists x\in X$ s.t. $(x,y)\in f$. A function $f: X\longrightarrow Y$ is said to be one-to-one and onto (or bijective) if it is both one-to-one and onto (or both injective and surjective).

Let $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ be two functions. Then the composition or the composite function $g\circ f: X\longrightarrow Z$ is defined by $g\circ f(x)=g(f(x))$ $\forall x\in X$. The function composition $\circ$ may be considered as an operation and it is associative.

Lemma. If $h: X\longleftrightarrow Y$, $g:Y\longleftrightarrow Z$ and $f:Z\longleftrightarrow W$, then $f\circ(g\circ h)=(f\circ g)\circ h$.

Note that $\circ$ is not commutative i.e. it is not necessarily true that $f\circ g=g\circ f$ even when both $f\circ g$ and $g\circ f$ are defined.

The following lemmas will be useful when we study group theory later.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are one-to-one, so is $g\circ f: X\longrightarrow Z$.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are onto, so is $g\circ f: X\longrightarrow Z$.

As an immediate consequence of combining these two lemmas, we obtain

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are bijective, so is $g\circ f: X\longrightarrow Z$.

If $f\subset X\times Y$, then the inverse graph $f^{-1}\subset Y\times X$ is defined by
$$f^{-1}=\{(y,x)\in Y\times X: (x,y)\in f\}.$$
If $f: X\longrightarrow Y$ is one-to-one and onto (bijective) then its inverse graph $f^{-1}$ is a function $f^{-1}: Y\longrightarrow X$. The inverse $f^{-1}$ is also one-to-one and onto.

Lemma. If $f: X\longrightarrow Y$ is a bijection, then $f\circ f^{-1}=\imath_Y$ and $f^{-1}\circ f=\imath_X$, where $\imath_X$ and $\imath_Y$ are the identity mappings of $X$ and $Y$, respectively.

Let $A(X)$ be the set of all one-to-one functions of $X$ onto $X$ itself. Then $(A(X),\circ)$ is a group. If $X$ is a finite set of $n$-elements (we may conveniently say $X=\{1,2,\cdots,n\})$, then $(A(X),\circ)$ is a finite group of order $n!$, called the symmetric group of degree $n$. The symmetric group of degree $n$ is denoted by $S_n$ and the elements of $S_n$ are called permutations.

This is the first of a series of lecture notes on group theory I intend to write for undergraduate Modern Algebra I course I am teaching in the fall semester. Before we begin to discuss the 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 $f$. $\underline{A}$ 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 decieved 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 $a,b\in G$. If you have taken linear algebra (and I believe you have), you would notice that a linear map is a map that preserves vector addition and scalar multiplication. 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 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 a 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 groups (symmetry groups). The most general symmetry group of finite order is called a symmetric group. In mathematics, the embedding theorem is conceptually and philosophically important (though 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.

Remark. There is a mathematical structure called algebras over field $K$ (usually $K=\mathbb{R}$ or $K=\mathbb{C}$). An algebra $\mathcal{A}$ over field $K$ is a vector space over $K$ with a product $\cdot:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ which is distributive over addition:
$$a(b+c)=ab+ac,\ (a+b)c=ac+bc,\ \forall\ a,b,c\in\mathcal{A}.$$
(Here, the symbol $\forall$ is a logical symbol which has meaning “for each”, “for any”, “for every”, or “for all” depending on the context. I will talk more about logical symbols next time as I will use them often.) Note that an algebra $\mathcal{A}$ over field $K$ is not an algebra because the scalar product is not an operation on $\mathcal{A}$. The scalar product is in fact an action of the multiplicative group $K\setminus\{0\}$ on $\mathcal{A}$. Algebras over field $K$ are important structures in functional analysis.