# Group Dictionary

A group is a set of elements $\{g\}$ and an associated binary operation $\circ$, such that given any two elements $g,g'\in G,\;g\circ g' \in G$; this is a property called closure. In particular, a group is defined by a series of axioms. It must:

1. Have an identity element $\mathbb{I}:\mathbb{I}g=g\mathbb{I} \forall g\in G$

2. Every element has an inverse $\forall g \in G \exists g^{-1} : g\circ g^{-1} = \mathbb{I}$
Sidenote: The 'right' inverse $g^{-1}\circ g =\mathbb{I}$ and 'left' inverse $g\circ g^{-1}=\mathbb{I}$are distinct. If they are equal, the the element $g$ is said to be invertible.

3. The elements are closed under $\circ$

4. Associativity $\left(g\circ g'\right)\circ g'' = g\circ\left(g'\circ g''\right)$

We denote the order of the group, the number of elements in the group, as $\vert G\vert$. This page will outline other properties and terminology about groups and group theory. The group operation can be any binary operation that is associative. This is typically referred to and denoted by multiplication, which we will use throughout these notes, though isn't restricted to multiplication. If the operation is also commutative, then we say that the group is abelian, otherwise it is non-abelian.

There is only a single group operation which uniquely defines a given group. This can be prooved from the cancellation law, namely: which can be proved in both cases by left or right multiplication with the inverse $a^{-1}$.

## Common Groups

There are two main groups that come up often in the study of group theory. Their group operations are common binary operations in mathematics: matrix multiplication and function composition.

1. The General Linear Group$GL_{n}$:
This is defined as the group of $n\times n$ matrices combined under matrix multiplication.

2. The Symmetric Group$S_{n}$:
Given a function on a set $T$, $f:T\rightarrow T$, $f$ is said to be a permutation if it is bijective. Permutations of $T$ forma group under permutation, and we denote by $S_{n}$ the group of permutations on a set of order $n$.

The group $S_{3}$ is of particular interest because it is the smallest non-abelian group. If we define two elements, the cyclic permutation $x:\{1,2,3\}\rightarrow\{3,1,2\}$, and the transposition $y:\{1,2,3\}\rightarrow\{2,1,3\}$, then the group has and we have elements All combinations of $x,y$ can be shown to be equal to the elements above using the multiplication rules given above that. These are called defining rules of the group.

What makes these groups especially interesting is that they other groups are often subgroups of these larger groups.

## Subgroup

A subgroup of a group $G$ is defined as a subset $H\subseteq G$ such that $H$ is also a group; it is closed and each element has an inverse. A proper subgroup is defined as a non trivial subgroup i.e $H\neq G$.

### Maximal Subgroup

A maximal subgroup is defined, in group theory, is a proper subgroup $H$ of $G$ such that no other proper subgroup of G contains $H.$

## Cyclic Groups

These are an important category of subgroups generated by an element of the group $x\in G$. The cyclic subgroup of $x$ is defined as This is the smallest subgroup of $G$ that contains $H$, and is often denoted as the group generated by $x,\;\langle x\rangle$.

The order of the cyclic group is also referred to as the order of the element $x$ that generates the group; this can in fact be infinite if $x^{n}\neq \mathbb{I}\forall n$. The identity element has order 1, and is the only element with order 1. In fact, we can show the following property of cyclic groups

• Denote by $S$ the set of integers $k\in\mathbb{Z}:x^{k}=1$.

In this case $S\subseteq \mathbb{Z}^{+}$ the group of integers under addition. This we can see as $x^{r+s}=\mathbb{I}\mathbb{I}=\mathbb{I},\;\implies r+s\in S$. We can also show obviosuly that if $x^{k}=\mathbb{I},\;x^{-k}=\mathbb{I}^{-1}=\mathbb{I}\implies -k\in S$.

• Two powers $x^{r}=x^{s},r\geq s$, are equal iff $r-s=1 \implies r-s\in S$

• Let $S$ be a proper subgroup. Then $S=\mathbb{Z}_{n}$ such that $x^{n}=\mathbb{I}\implies \langle x\rangle=\{x^{k}:k\in\mathbb{Z}_{n}\}$, where $n$ is the smallest positive integer in $S$.

Consider also the case where $x$ is an element with finite order in a group and take an integer $k=nq+r:\,q,r\in\mathbb{Z},\,0\leq r. We thus have

• $x^{k}=x^{r}$
• $x^{k}=1$ iff $r=0$
• For $d$ greatest common divisior of $k$ and $n$, the order of $x^{k}$ is $n/d$.

### Generating Sets

The subgrup of $G$ that can be generated by a subset $U$ is defiend as the smallst subgroup $H:U\in H$, and all elements of $H$ can be expressed as a combination of the elements in $U$ and their inverses. We call $U$ the generator of the group $H$.

## Homomorphisms

iven a pair of groups $G,G'$, then a homomorphism $\phi:G\rightarrow G'$ is a function that maps an element of $G$ to an element of $G'$, such that $forall a,b\in G$ i.e. the map respects the law of composition of the two groups. Important examples og Homomorphisms include

1. The determinant $\text{det}:GL_{n}(\mathbb{R})\rightarrow\mathbb{R}^{\times}$
2. The sign homomorphism $\sigma:S_{n}\rightarrow \pm 1$
3. The exponential map $e:\mathbb{R}^{+}\rightarrow\mathbb{R}^{\times}$
4. The map $\phi:\mathbb{Z}^{+}\rightarrow G:\,\phi(n)=a^{n}$ for an element $a\in G$
5. The absolute value map $\mathbb{C}^{\times}\rightarrow \mathbb{R}^{\times}$

We can also define two types of 'useless' homomorphism, the trivial homomrphism $\phi:G\rightarrow G':\phi(g)=\mathbb{I}_{G'}\forall g\in G$, and the inclusion map $i:H\rightarrow G :\,i(x)=x\forall x\in H$. We also note that as a consequence on its definition, $\phi(\mathbb{I}_{G})=\mathbb{I}_{G'}$ and $\phi(a^{-1})=\phi(a)^{-1}$.

Related to the homomorphism, we can define two important subgroups the image and the kernel. We define the image $\text{im}\phi=\{x\in G':\,x=\phi(a),\,a\in G\}=\phi(G)$, and the kernal $\text{ker}\phi=\{a\in G :\,\phi(a)=\mathbb{I}_{G'}$. As interesting examples, we can se that the image of the 4th example above is the cyclic subgroup $\langle a\rangle$. We can also show tht the kernal of the determinant map is the special linear group, matrices with determinant 1, and that the 'Alternating' group is the image of the sign map.

The kernel can also be used because it can tell us which pairs of elements have the same image in G'. In particular, consider a coset of a subgroup $H$. AS with inverses, we distinguish between the left and right cosets, but generally they are given as $aH=\{a\circ h\forall h\in H, a\in G\setminus H$, that is $a$ is an element of $G$ not in the subgroup. Now consider $a,b\in G:\phi(a)=\phi(b)$. This is equivalent to stating that $a^{-1}b\in\text{ker}\phi$, which in turn means $b\in aK$ and $bK=aK$. This can be extended to prove that a homomorphism is injective iff $\text{ker}\phi=\{\mathbb{I}_{G}\}$.

The kernel can also be relate to a group operation called conjgution. Specifically, the conjugate of $a$ with respect to $g$ is defined as $gag^{-1}$. A subgroupo $N\subseteq G$ is called normal if $gNg^{-1}\in N\forall g \in G$. We can then show that the kernel of a homomorphism is a normal subgroup. Given any $a\in K(\phi)$, then $\phi(gag^{-1}=\phi(g)\mathbb{I}\phi(g')=1$ and the conjugate of $a$ is also in the kernel. This in tern implies the special linear group is a normal subgroup of the linear, and the alternating group is a normal subgroup of the permutation group. Subgroups of any abelian group are normal by definition, but this is not the case for non-abelian groups. For example, the cyclic group of $y$ in $S_{3}$ is not. Related to conjugation, we can define a new subgroup called the cetnre $Z$ of $G$, which is given by the set of elements that commute with all elements of $G$. It is definitionally a normal subgroup.

### Isomorphisms

These are defined as morphisms which are bijective, and respect the group conjugation rules. To check that a morhpism is an isomorphism, we require that $\text{im}\phi=G'$, and that $\text{ker}\phi=\mathbb{I}_{G}$. That is to say, the map is surjective and injective. Because the map is biejctive, we can define a corresponding inverse map $\phi^{-1}:G'\rightarrow G$ that is also an isomprohism, which can be simply shown. We require $\forall x,y\in G', \phi^{-1}(xy)=\phi^{-1}(x)\phi^{-1}(y)$, which we can write as $c=ab \text{for} a,b,c\in G$. As $phi$ is bijective, we need to show that $\phi(ab)=\phi(c)$, we we can show as $\phi(a)\phi(b)=xy$. This suggests that the two groups behave identically under the group law. Isomorphism between groups is denoted $G\approx G'$. Often when discussing isomorphic groups we freely switch between which representation we use as they have the same group structure.

All the groups isomporphic to a group $G$ are called the isomorphism class of $G$. An important example of an isomporphism class are all groups of prime order p, which are all cyclic. There are also two isomorphism classes of groups order 4 and 5 classes of group order 12.

### Automorphisms

These are a special class of isomorphism defined as a map $\phi:G\rightarrow G$, that is they map a group to itself. A particularly important automorphism is the conjugation map. This is trivial for abelian groups, but not for non-abelian groups. Because the conjugate of an element, $x'=gxg^{-1}$, is the image of $x$ under an automorphism, it follows that the conjugate has similar properties e.g. the same order as the element $x$.

### Group commutation

Related to determining if two elements are conjugate is the relation $yg=gx\implies y=gxg^{-1}$. We can define an operation called the commutator of two group elements $aba^{-1}b^{-1}$, such that $aba^{-1}=b\implies ab=ba\implies aba^{-1}b^{-1}=1$.

## Equivalence Relations and Partitions

Paritioning a set means to subdivide it into a collection of disjoint, nonempty subsets $S_{i}:S=\cup_{i}S_{i}$. An equivalence relation is a relation that holds between pairs of elements of $S$, and can be written $a\sim b$. These relations are transitive, $a\sim b$ and $b\sim c\implies a\sim c$, symmetruc $a\sim b\implies b\sim a$, and reflexive $a\sim a\forall a$. Conjugacy is a good example of such an equivalence relation. Suppose $a\sim b \implies b=g_{1}ag_{1}^{-1}$, and $b\sim c\implies c=g_{2}bg_{2}^{-1}$. In this case,$c=g_{2}g_{1}ag_{1}^{-1}g_{2}^{-1}\implies a\sim c$. Partitioning sets usually relies on defining an equivalence relation between items in each subset. In fact, given an equivalence relation on $S$ the equivalence classes of the elements define a partiion of S.

Given a partition of a set $S$, we can define a related set $\bar{S}$ whose elements are the partitions of $S$. We denote a subset $U\subseteq S$ as $[U]$ when we are referring to the corresponding element of $\bar{S}$. For any equivalence relation, we can define a surjective map $\pi:S\rightarrow\bar{S}$, which maps an element of $S$ to its equivalence class. We can also denote $\pi(a)$ as $\bar{a}$. One disadvantage of this notation is that there are multiple elements in the same equivalence class, so it can be helpful to define a representative element.

### Equivalence Relations and Maps

In fact, any map $f:S\rightarrow T$ defines an equivalence relation on the domain $S$. We can define the inverse image of an element $t\in T$ as the set $\{s\in S:f(s)=t\}$. In general, the inverse image cannot be written $f^{-1}(t)$ unless $f$ is bijective, in which case the inverse image is trivial. These are also referred to as the fibres of the map $f$, and non-empty fibres are the equivalence classes of $S$ under $f^{-1}$. The equivalence classes of $S$ can in turn be seen as the image $f(S)$.

The equivalence relation defined by a group homomorphism is usually denoted $a\equiv b\implies \phi(a)=\phi(b)$, and is referred to by the term congruence. Thus, elements are congruent if and only if $b$ is in the coset $aK$ of the kernel $K$. We can in fact show that the fibre of $\phi$ that contains an element $a$ is equivalent to the coset $aK$. These cosets partition the group, and correspond to the elements of the iamge of $\phi$.

## Cosets

We have already discussed cosets above, but to summarise given a subgroup $H\subseteq G$ and an element $a\in G\setminus H$, the left and right cosets of $a$ are given by $aH$ and $Ha$ respectively. Cosets form the equivalence classes of the congruence operation, namely $a\equiv b$ if $b=ah$ for some $h\in H$. Thus, the left cosets of a subgroup $H$ partition the group $G$. The number of left cosets in a group is called the index of $H$ in $G$ and is denoted $[H:G]$.

We can show that all cosets of a group have the same order, as they are all produced from a map $a\rightarrow aH$ that is invertible. Fro this, we can define the Counting Formula, namely that $\vert G\vert =[H:G]\vert H\vert$. This implies for example that $[H:G]=\vert G\vert /\vert H\vert$. This leads to Lagrange's theorem, namely that the order of a subgroup $H$ divides the order of the group $G$. This in turn implies the same about the order of the elements of a finite order group, because the corresponding cyclic group is a subgroup of $G$. This in turn is what implies that any group of prime order is a cyclic group.

We can also use the counting formula in the case of a group homomorphism using the cosets of the kernel. In this case, we have $[G:\text{ker}\phi]=\vert \text{im}\phi\vert$, and $\vert G\vert =\vert K\vert\vert \text{im}\phi\vert$. This has the consequence that $|K|$ divides $|G|$ and $|\text{im}\phi|$ divides $|G|$ and $|G'|$.

The index has a multiplicative rule given a sequence of subgroups. In particular, given $J\subset H\subset G$, then $[G:K]=[G:H][G:K]$.

### Right Cosets

The above discussions, including the congruence relation, were defiend using the left cosets of the subgroup $H\subset G$. We can in turn define the right cosets, $\{Ha\forall a\in G\setminus H\}$. We thus have the right congruence relation $a\equiv b\;\text{iff}\,b=ha$ for some $h\in H$. These are distinct from the left cosets unless the subgroup is normal. The following propositions are equivalent

1. $H$ is a normal subgroup such that $ghg^{-1}\in H,\forall\, h\in H,\,g\in G$
2. $\forall\,g \in G,\,gHg^{-1}=H$
3. $\forall\,g\in G,\,gH=Hg$
4. Every left coset of $H$ is also a right coset

We can prove the third statement by noting that cosets partition the group, and thus if $gH$ is equal to any other coset it must be $Hg$.

We can also show that, given a subgroup $H$ of $G$ and an element $g\in G$, then the conjugate of $H$ is also a subgroup. If $G$ has only one subgroup of order $r$, then this group is normal. These properties follow from the fact that conjugation is an automorphism.

## The Correspondence Theorem

Consider a group homomorphism $\phi:G\rightarrow\mathcal{G}$, and $H\subset G$. We define the restriction of $\phi$ to the domain $H$ as $\phi|_{H}:H\rightarrow\mathcal{G}$. The kernel of this restricted homomorphism is given by the intersection $\text{ker}\phi\cap H$, and it's image $\text{im}\phi|_{H}=\phi(H)$. From the counting formila, we know that the image of $\phi|_{H}$ divides $|H|$ and $|\mathcal{G}|$. If they have no common factor, then the image of $H$ must be the idenity and $H$ is entirely contained in the kernel.

An example, consider restricting the sign homomorphism $\sigma$ to a subgroup of $S_{n}$. $\text{im}\sigma$ has order 2, and thus any subgroup with odd order is contained in the kernel of $\sigma$. This is true when $H$ is a cyclic subgroup generated by a permutation with odd order in the group, which is thus an even permutation. A permutation with even order can be even or odd.

Now, consider consider $\phi:G\rightarrow\mathcal{G}$ with kernel $K$, and $\mathcal{H}\subset\mathcal{G},\,H=\phi^{-1}(\mathcal{H})$. In this case, $H\subset G,\,K\in H$. If $\mathcal{H}$ is a normal subgroup, then so is $H$. If $\phi$ is surjective and $H$ is a normal subgroup, then so is $\mathcal{H}$. A consequence of this proposition is, for example, that the Special Lienar group is a normal subgroup of $GL_{n}$, which we can see as the determinism map as it's range in $\mathbb{R}^{\times}$, an abelian group.

The proof of this proposition requires we remember $\phi^{-1}$ is not a map, but the set of elements in $G$ taken to the range of $\phi$ we are acting on. We know that $\phi(x)=\mathbb{I}_{\mathcal{G}}\forall\,x\in K$. As it is a subgroup, $\mathbb{I}_{\mathcal{G}}\in\mathcal{H}$, and thus $H=\phi^{-1}(\mathcal{H})$ contains the kernel. We can further show that, due to the properties of the homomorphism, $H$ is a subgroup.

In turn, if $\mathcal{H}$ is normal, then for any given $g\in G,\,x\in H$, $\phi(gxg^{-1})=\phi(g)\phi(x)\phi(g)^{-1}$ is a conjugate of $\phi(x)$ and is contained in $\mathcal{H}$. This in turn implies $gxg^{-1}\in H$, and thus $H$ is normal. Consider the other direction: for a surjective $phi$, given $a\in \mathcal{H},\,b\in \mathcal{G}$, we know that $\exists x\in H,\,y\in G:\,\phi(x)=a,\,\phi(y)=b$. As $yxy^{-1}\in H$, we know that $\phi(yxy^{-1})=bab^{-1}\implies \mathcal{H}$ is a normal subgroup.

This leads to the correspondence theorem: given a surjective group homomorphism $\phi:G\rightarrow\mathcal{G}$, there is a bijective correspondence beteen subgroups of $\mathcal{G}$ and subgroups of $G$ that contain $\text{ker}\phi$. If $H$ and $\mathcal{H}$ are corresponding subgroups, then one is normal iff the other is normal, and $|H|=|\mathcal{H}||K|$.

## Product Groups

Consider a pair of groups $G,G'$. The product set $G\times G'$ with pairs of elements $(a,a')$ can be made in to a group by defining a cartesian-productified group product

The group under this operation is called the product of $G$ and $G'$. It is related to $G$ and $G'$ by a set of 4 homomorphisms, which we can define as:

It is always interesting to determine a product decomposition of a group $G$, such that $H\times H'\approx G$. The factors will have a simpler structure generally than the wider group. This is not always possible, but there are some examples. E.g consider a cyclic group of order 6 $C_{6}$, which is in fact isomorphic to a product $C_{3}\times C_{2}$.This is in fact generally true for any cyclic group of order $rs$, such that $r$ and $s$ have no common factor (are relatively prime).

We can define product groups by examining the following propositions. Consider two subgroups $H,K\subset G$, and define the multiplicative map $f:H\times K\rightarrow G$ define by $f(h,k)=hk\forall h\in H,\,k\in K$. We thus have $\text{im}f=HK=\{hk\forall h\in H,\,k\in K\}$. We thus have that

1. $f$ is injective iff $H\cap K=\mathbb{I}_{G}$
2. $f$ is a homomorphism from the product group $H\times K$ to $G$ iff elements of $K$ commute with $H:hk=kh$
3. If $H$ is a normal subroup, then HK is a subgroup of $G$
4. $f$ is an isomoprhism from $H\times K$ to $G$ iff $H\cap K=\mathbb{I}_{G}$, $HK=G$, and $H,K$ are normal subgroups of $G$.

We note that the multiplication map can be bijective without being a group homomorphism, as is the case for $S_{3}$ with subgroups $\langle x\rangle$ and $\langle y\rangle$.

Now consider the proof of the above propositions. Firstly, if $H\cap K$ contains a non-trivial element $x$, then $x^{-1}\in H$ and thus $f(x^{-1},x)=1=f(1,1)$, and $f$ is not an injective map. If instead the intersection is only $\{1\}$, then if $h_{1}k_{1}=h_{2}k_{2}$, we can left-multiply and right-multiply by $h_{1}^{-1}$ and $k_{2}^{-1}$ respectively to obtain $k_{1}k_{2}^{-1}=h_{1}^{-1}h_{2}$, which as the left side is in $K$ and the right in $H$ means they must both equal idenity. hus, $k_{1}=k_{2},\,h_{1}=h_{2}$.

Secondly, given two pairs in the product group $(h_{1,2},k_{1,2})$, their product is $(h_{1}h_{2},k_{1}k_{2})$, and $f(h_{1}h_{2},k_{1}k_{2})=h_{1}h_{2}k_{1}k_{2}$. However, $f(h_{1},k_{1})f(h_{2}k_{2})=h_{1}k_{1}h_{2}k_{2}$ and thus these are equal iff $h_{2}k_{1}=k_{1}h_{2}$. Relatedly, let $H$ be a normal subgroup. Then we note that $KH=\{kH\}\cup K$, and $HK=K\cup\{Hk\}$. As $H$ is normal $kH=Hk$. We can further show $HK$ is closed under multiplication and inverses, and this $HK=KH$.

Finally, suppose $H$ and $K$ are normal subgroups that satisfy $HK=G$. We need $hk=kh\forall h\in H,\,k\in K$. Given the commutator and under associativty we can write $h(kh^{-1}k^{-1})=(hkh^{-1})k^{-1}$. As these are normal subgroups, the left side is in $H$ and the right side is in $K$, and thus as $H\cap K=\{1\}$ this means the commutator must equal idenity, giving $hk=kh$.

As an example, we can use this to show their are two isomorphism classes of group order $4$: the order-4 cyclic groups, and the groups isomorphic to $C_{2}\times C_{2}$.