Chapter 1: Groups

Monoids

A monoid is a set with an associative law of composition and a unit element.

An abelian monoid is a monoid with a commutative law of composition.

A homomorphism of a monoid $M$ into a monoid $M'$ is a mapping $f:M\to M'$ such that $f(xy)=f(x)f(y)$ for all $x,y\in M$ , and $f(e)=e'$ where $e'$ is the unit element of $M'$ .

If $M=M'$ , then a homomorphism is called an endomorphism.
If a homomorphism is injective, then it is called an embedding (denoted as $f:M\hookrightarrow M'$ ).
The kernel $\text{Ker}f$ of a homomorphism $f:M\to M'$ is the set $\{x\in M|f(x)=e'\}$ . The kernel is a submonoid.
The image $\text{Im}f$ of a homomorphism $f:M\to M'$ is the set $\{x'\in M'|x'=f(x)\text{ for some }x\in M\}$ . The image is a submonoid.
The composition of homomorphisms is a homomorphism.

A homomorphism is an isomorphism if it is bijective. In this case we write $M\approx M'$ .

If $M=M'$ , then an isomorphism is called an automorphism.
The composition of isomorphisms is an isomorphism.

Groups

A group is a monoid such that every element has an inverse.

If in a set with an associative law of composition there is a left unit element and each element has a left inverse, then the set is a group, and these are also the unit element and inverses of elements.
The kernel and image of a group-homomorphism are subgroups of the domain and range groups, respectively.
The kernel is trivial iff the homomorphism is injective (the direct implication is not true for monoids).

A subset $S\subset G$ is a set of generators for $G$ if every element of $G$ can be expressed as a (finite) product of elements of $S$ and their inverses. We write $G=\langle S\rangle.$

A cyclic group is generated by a single element. Such an element is called a cyclic generator of the group.
If a map $S\to G'$ defined on a set of generators $S\subset G$ can be extended to a homomorphism $G\to G'$ , then such an extension is unique.

The direct product of groups $(G_{i},\cdot)$ , $i\in I$ , is $(\prod_{i\in I}G_{i},\cdot)$ , where $(x_{i})_{i\in I}\cdot(y_{i})_{i\in I}=(x_{i}\cdot y_{i})_{i\in I}.$

If for $k=1,\dotsc,n$ , $H_{k}$ is a subgroup of $G$ , $H_{k}\cap H_{1}\dotsb H_{k-1}=\{e\}$ , for all $i<k$ and all $x\in H_{k}$ and $y\in H_{i}$ , $xy=yx$ , and $G=H_{1}\dotsb H_{n}$ , then $f:\prod_{k=1}^{n}H_{k}\to G$ given by $f((x_{k})_{k=1}^{n})=x_{1}\dotsb x_{n}$ is an isomorphism.

A left (right) coset of a subgroup $H$ of $G$ is $aH=\{a\}H$ for some $a\in G$ .

Left (right) cosets form a partition of $G$ . The number of cosets is the index $(G:H)$ of $H$ in $G$ . The index $(G:1)$ of the trivial subgroup of $G$ is called the order of $G$ .
If $H'\subset H\subset G$ are subgroups, then $(G:H)(H:H')=(G:H')$ .
- In particular, $(G:H)(H:1)=(G:1)$ .
- Each group of a prime order is cyclic.

Normal subgroups

A subgroup $N$ of $G$ is normal ( $N\trianglelefteq G$ ) if $xNx^{-1}\subset N$ for all $x\in G$ .

The intersection of (any collection of) normal subgroups is a normal subgroup.
$N$ is a normal subgroup iff it is the kernel of a homomorphism.
The factor group of $G$ by $N$ , $G/N$ ( $G$ modulo $N$ ), is the group of the cosets of $N$ . We say $x\equiv y\,\mod N$ ( $x$ and $y$ are congruent modulo $H$ ) iff $x$ and $y$ are in the same coset of $N$ .

The normalizer of a subset $S\subset G$ is the set $N_{S}=\{x\in G|xSx^{-1}=S\}$ .

$N_{S}$ is a subgroup of $G$ , but not necessarily normal.
If $H$ is a subgroup of $G$ , then
- $H\subset N_{H}$ ,
- $N_{H}$ is the largest subgroup of $G$ containing $H$ as a normal subgroup, and
- if $H'$ is a subgroup of $N_{H}$ then $HH'=H'H$ is a group, and $H\trianglelefteq HH'$ .

The centralizer of a subset $S\subset G$ is the set $Z_{S}=\{x\in G|xyx^{-1}=y,y\in S\}$ .

$Z_{S}$ is a subgroup of $G$ , but not necessarily normal.
$Z_{G}$ is called the center of $G$ , and it is a normal subgroup of $G$ .
$Z_{S}$ is a normal subgroup of $N_{S}$ . Indeed, for any $z\in N_{S}$ , for every $y\in S$ , $yz=zy'$ for some $y'\in S$ , and for every $x\in Z_{S}$ , $zxz^{-1}yzx^{-1}z^{-1}=$ $zxy'x^{-1}z^{-1}=$ $zy'z^{-1}=$ $y$ .

The sequence of homomorphisms $G'\overset{f}{\to}G\overset{g}{\to}G''$ is exact if $\text{Im}f=\text{Ker}g$ .

$0\to G'\overset{f}{\to}G\overset{g}{\to}G''\to0$ is exact means, in particular, that $f$ is injective and $g$ is surjective.

Canonical homomorphisms

Let $N$ be a normal subgroup of $G$ . Then $\phi:G\to G/N$ given by $\phi(x)=xN$ is the canonical map.
Let $f:G\to G'$ be a homomorphism, $\text{Ker}f=K$ , and $\phi:G\to G/K$ be the canonical map. Then, there is a unique homomorphism $f_{*}:G/K\to G'$ such that $f_{*}\circ\phi=f$ , namely $f_{*}(xK)=f(x)$ . And further, $f_{*}$ is injective, i.e. $f_{*}:G/K\hookrightarrow G'$ is an embedding. We say that $f_{*}$ is induced by $f$ .
- In fact, $f_{*}=j\circ\lambda$ where $\lambda:G/K\to\text{Im}f_{*}=\text{Im}f$ is an isomorphism, and $j:\text{Im}f\to G'$ is the inclusion map. Hence, $f=j\circ\lambda\circ\phi$ .
Below $\hat{g}$ means the restriction of the range of $g$ to its image.

$\begin{CD}0@>>>K@>\text{inc}>>G@>\hat{f}>>f(G)@>>>0\\ @.@VV\text{can}V@VV\phi V@|@.\\ 0@>>>0@>>>G/K@>\hat{f}_{*}=\lambda>>f(G)@>>>0 \end{CD}$
Let $H\subset K=\text{Ker}f\subset G$ where $f:G\to G'$ is a homomorphism, $N$ the smallest normal subgroup of $G$ containing $H$ ( $H\subset N\subset K$ ), and $\phi:G\to G/N$ the canonical map. Then, there is a unique homomorphism $f_{*}:G/N\to G'$ induced by $f$ such that $f_{*}\circ\phi=f$ , namely, $f_{*}(xN)=f(x)$ .
Below $\hat{g}$ means the restriction of the range of $g$ to its image.

$\begin{CD}0@>>>K@>\text{inc}>>G@>\hat{f}>>f(G)@>>>0\\ @.@VV\text{can}V@VV\phi V@|@.\\ 0@>>>K/N@>\text{inc}>>G/N@>\hat{f}_{*}>>f(G)@>>>0 \end{CD}$
Let $N\subset K\subset G$ be normal subgroups. Then, there is a surjective homomorphism $f:G/N\to G/K$ defined by $f(xN)=xK$ with the kernel $\{xN|x\in K\}$ . Therefore, we have a canonical isomorphism $(G/N)/(K/N)\approx G/K$ .

$\begin{CD}0@>>>K@>\text{inc}>>G@>\text{can}>>G/K@>>>0\\ @.@VV\text{can}V@VV\text{can}V@|@.\\ 0@>>>K/N@>\text{inc}>>G/N@>f>>G/K@>>>0\\ @.@VV\text{can}V@VV\text{can}V@|@.\\ 0@>>>0@>>>(G/N)/(K/N)@>f_{*}=\lambda>>G/K@>>>0 \end{CD}$
Let $H,H'\subset G$ be subgroups, and $H'\subset N_{H}$ . Then, $H\cap H'\trianglelefteq H'$ (and as before $H\trianglelefteq HH'=H'H\le G$ ). Then, there is a surjective homomorphism $f:H'\to H'H/H$ defined by $f(h')=h'H$ with the kernel $H\cap H'$ . Therefore, we have a canonical isomorphism $H'/(H\cap H')\approx HH'/H$ .

$\begin{CD}0@>>>H@>\text{inc}>>HH'@>\text{can}>>HH'/H@>>>0\\ @.@VV|_{H_{1}}V@VV|_{H_{1}}V@|@.\\ 0@>>>H\cap H'@>\text{inc}>>H'@>f>>HH'/H@>>>0\\ @.@VV\text{can}V@VV\text{can}V@|@.\\ 0@>>>0@>>>H'/(H\cap H')@>f_{*}=\lambda>>HH'/H@>>>0 \end{CD}$
Let $f:G\to G'$ be a homomorphism, and $H'\trianglelefteq G'$ . Then, $H=f^{-1}(H')\trianglelefteq G$ is the kernel of the composite of $f$ and the canonical map $\phi:G'\to G'/H'$ . Therefore, there is an injective canonical homomorphism from $G/H$ to $G'/H'$ , which is an isomorphism if $f$ is surjective.

$\begin{CD}0@>>>H@>\text{inc}>>G@>\text{can}>>G/H@>>>0\\ @.@VVf|_{H}V@VVfV@VV(\phi\circ f)/HV@.\\ 0@>>>H'@>\text{inc}>>G'@>\phi>>G'/H'@>>>0 \end{CD}$