## Quick Review of Group Theory

### Key definitions

**Definition:** A group is a set \(G\) together with a map \(m: G\times G\to G\) satisfying the following axioms:

- There is an element $e\in G$ such that $m(e,x)=m(x,e)=x$ for all $x\in G$.
- For all $x,y,z\in G$, we have $m(x,m(y,z))=m(m(x,y),z)$.
- For all $x\in G$, there is $y\in G$ such that $m(x,y)=m(y,x)=e$.

We usually just write $ab$ or $a+b$ for $m(a,b)$; and we usually write $G$, rather than $(G,m)$ when speaking about a group.

One can weaken these axioms in various ways and obtain an equivalent definition.

For any group $G$ and $x\in G$:

- there is only one element $e$ satisfying axiom (1).
- the regrouping in axiom (2) extends to arbitrary many elements, so the product $a_1 a_2 \cdots a_n$ is well defined for any set of elements $a_1,a_2,\ldots, a_n\in G$.
- the inverse $y$ for $x$ required by axiom (3) is unique.

**Definition:** If $G$ is a group and $ab=ba$ for all $a,b\in G$ then $G$ is called *abelian*.

**Definition:** If $G$ is a group and $g\in G$ then the *order* of $g$ is the smallest positive integer $n$ such that $g^{n}=e$ (or infinity, if no such $n$ exists).

**Definition:** If $G$ is a group, and $H$ is a nonempty subset of $G$ which is a group when the multiplication of $G$ is restricted to $H$, then $H$ is called a *subgroup* of $G$. $H$ is a subgroup if:

- for all $a,b\in H$, $ab\in H$,
- if $a\in H$, then $a^{-1}\in H$.

**Definition:** If $G$ and $H$ are groups, and $f:G\to H$ is a function, then $f$ is called a *homomorphism* if $f(ab)=f(a)f(b)$ for all $a,b\in G$ and and *isomorphism* if it is a bijective homomorphism. The *kernel* of a homomorphism $f$ is the subgroup of $G$ consisting of elements $g$ such that $f(g)=e$. The *image* of a homomorphism $f$ is the subgroup of $H$ consisting of elements $h\in H$ such that $h=f(g)$ for some $g\in G$.

**Definition:** If $G$ is a group and $X$ is a set, then a map $m:G\times X\to X$ is called a (left) action of $G$ on $X$ if $m(e,x)=x$ for all $x\in X$ and $m(a,m(b,x))=m(ab,x)$ for all $a,b\in G$ and $x\in X$. We usually write $ax$ for $m(a,x)$.

### Examples

- The integers, rational numbers, real numbers, and complex numbers are all groups under addition.
- The nonzero rational numbers, real numbers, and complex numbers are all groups under multiplication.
- For $n>0$, the set $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ are a group under addition.
- The subset of $\mathbb{Z}/n\mathbb{Z}$ consisting of elements $a$ that are invertible (i.e. such that the congruence $ax\equiv 1\pmod{n}$ has a solution) form a group under multiplication. This group is called $(\mathbb{Z}/n\mathbb{Z})^{\times}$. If $n=p$ is prime, then $(\mathbb{Z}/p\mathbb{Z})^{*}$ consists of the $p-1$ nonzero congruence classes.
- The invertible $n\times n$ matrices $\mathrm{GL}_{n}(F)$ where $F$ is any of $\mathbb{Q},\mathbb{R},\mathbb{C}$ form a group under matrix multiplication. These groups come with actions on $F^{n}$ given by matrix multiplication.
- For any set $X$, the set of bijective maps $X\to X$ form a group under composition of functions. This group is called the
*symmetric group*$S(X)$ on $X$. If there is a bijection from $X$ to $Y$, then $S(X)$ and $S(Y)$ are isomorphic. If $X=\{1,2,3,\ldots,n\}$ then $S(X)$ is usually written $S_n$ and is called the symmetric group on $n$ elements. Notice that $S(X)$ comes with an action on $X$ given by $(f,x)\mapsto f(x)$. - If $X$ is a regular $n-gon$ in the plane, the group of rigid motions $f$ of the plane such that $f(X)=X$ form a group under composition called the Dihedral group. Dummit and Foote call this group $D_{2n}$ since it has $2n$ elements, but others call it $D_{n}$ since it is the symmetries of an $n$-gon. The elements of $D_{2n}$ consist of \(\{1,r,r^2,\ldots, r^{n-1},s,sr,sr^2,\ldots, sr^{n-1}\}\) and the group law is determined by the rules $r^{i}r^{j}=r^{i+j}$ with exponents read modulo $n$, $s^2=1$, and $sr=r^{-1}s$. The group $D_{2n}$ comes with an action on the $n$ vertices of $X$ by $(f,v)\mapsto f(v)$ and on the $n$ sides of $v$ by $(f,s)=f(s)$.

## Cyclic Groups

**Definition:** A group $G$ is cyclic if there is an element $g\in G$ such that the homomorphism \(\phi_g:\mathbb{Z}\to G\) defined by $\phi_g(n)=g^{n}$ is surjective.

**Proposition:** Let $H\subset \mathbb{Z}$ be a propersubgroup. Then either $H={0}$ or there is a unique $n>0$ such that $H=n\mathbb{Z}$.

**Corollary:** A cyclic group is isomorphic either to $\mathbb{Z}$ or to $\mathbb{Z}/n\mathbb{Z}$ for some integer $n>0$.

**Properties of order:** Let $G$ be a group and $x\in G$.

- If $x^a$ has infinite order for some $a$, so do all nonzero powers of $x$.
- If $x^a=e$ and $x^b=e$ then $x^{\mathrm{gcd}(a,b)}=e$.
- If $x^a=e$ then the order of $x$ divides $a$.
- If $x$ has order $a$, then $x^k$ has order $a/\mathrm{gcd}(a,k)$.
- If $G$ is cyclic of order $n$ generated by $x$, then $x^a$ generates $G$ if and only if $\mathrm{gcd}(a,n)=1$.

**Proposition:** The subgroups of $G=\mathbb{Z}/n\mathbb{Z}$ are in bijection with the divisors of $n$. If $d$ is a divisor of $n$, then the unique subgroup of $G$ of order $d$ is generated by $n/d$.

## Euclidâ€™s Algorithm and Congruences

**Theorem:** Let $a$ and $b$ be nonzero integers. Then there exist integers $x$ and $y$ such that \(ax+by=d\) where $d$ is the greatest common divisor of $a$ and $b$.

**Theorem:** Let $n$ be a positive integer. The congruence equation

has solutions if and only if $d=\mathrm{gcd}(a,n)$ divides $b$. If this condition is satisfied, it has $d$ solutions of the form

\[x_0+k\frac{n}{d}\quad k=0,\ldots, d-1\]where $x_0$ is a representative for the unique solution to the congruence

\[\frac{a}{d}x_{0}\equiv \frac{b}{d}\pmod{\frac{n}{d}}.\]**Remark:** Notice that the congruence equation problem is equivalent to finding $x$ and $y$ so that \(ax+ny=b.\)