Groups

$$\newcommand{\G}{(G, *)} \newcommand{\Hom}{\text{Hom}} \newcommand{\Iso}{\cong} \newcommand{\Norm}{\unlhd } \newcommand{\Aut}{\text{Aut}} \newcommand{\Z}{\mathbb Z} \newcommand{\Ker}{\text{ker}} \newcommand{\Inn}{\text{Inn}} \newcommand{\Out}{\text{Out}} $$

1. Groups

Definition. Group

A group is a pair $\G$, where $G$ is a set, $∗ : G × G\to G$ a map called the group operation, such that:

• (associativity) $\forall a,b,c\in G$

  $$(a* b)* c = a* (b*c) $$

• (identity) There is an element $e\in G$, such that $\forall a\in G$

  $$e* a = a*e= a $$

• (inverse) $\forall a\in G$

  $$\exists b\in G.a*b=b*a=e $$

  Such $b$ is called inverse of $a$.

If $G$ has finite cardinality we call it a finite group, otherwise we call it an infinite group.

From now on, we write $ab$ to denote $a*b$

Example.

• Any linear space is a group under $+$, and linear transformations are group homomorphisms.
• The set of all bijections from a set $X$ to itself is called the permutation group $S_X$, with product defined to be function composition

Theorem. Basic facts of a group.

Let $\G$ be a group, then

1. (Cancellation Law) $\forall a,b,c\in G$

   $$ab=ac\implies b=c\\ ab=cb\implies a=c $$

3. The identity is unique.

4. The inverse is unique.

Definition. Commutative group
A group $\G$ is called commutative or abelian if $\forall a,b\in G.ab=ba$

Definition. Subgroup
Let $G$ be a group, $H\subset G$ is called a subgroup of $G$ (denoted by $H\leq G$) if

1. $H\neq \emptyset$ (this could also be replaced by $e\in H$)
2. $H$ is closed under inverse
3. $H$ is closed under product ($*$)

Lemma. Intersections of subgroups of a group is still a subgroup of this group.

Easy to prove by definition.

Definition. Homomorphism

$G$ and $H$ are two groups, $f:G\to H$ is called a group homomorphism if

$$\forall a,b\in G.f(ab)=f(a)f(b) $$

We denote the set of all homomorphisms from $G$ to $H$ by $\Hom(G, H)$

Lemma. a composition of group homomorphisms is still a group homomorphism

Definition. Endomorphism
A homomorphism from a group to itself is called an endomorphism.

Definition. Isomorphism
If $f\in\Hom(G, H)$ is a bijection, then $f$ is called a isomorphism from $G$ to $H$. If there’s a isomorphism between $G$ and $H$, we say $G$ and $H$ are isomorphic, denoted by $G\cong H$.

Definition. Automorphism group
Let $G$ be a group, the set of bijective group homomorphisms from $G$ to itself form a subgroup of $S_G$, called the automorphism group, denoted as $\Aut(G)$. Elements of this subgroup are called automorphisms.

Definition. Inner Automorphism
Let $G$ be a group, an automorphism $\sigma\in\Aut(G)$ is called an inner automorphism, if

$$\exists g.\sigma=x\mapsto gxg^{-1} $$

The set of all automorphisms from $G$ to itself is called the inner automorphism group of $G$, denoted by $\Inn(G)$

One can prove that $\Inn(G)\Norm \Aut(G)$ simply by definition.

Definition. Outer Automorphism
The quotient group $\Aut(G)/\Inn(G)$ is called the outer automorphism group of $G$.

Theorem. Basic facts about homomorphisms
Let $f: G\to H$ be a group homomorphism, then

1. $f(e_G)=e_H$
2. $\forall a\in G, f (a^{−1}) = (f (a))^{−1}$
3. $N\leq G\implies f(N)\leq H$
4. $M \leq H\implies f^{−1}(M ) \leq G$
5. $h \in\Hom(H, L)\implies h\circ f\in \Hom(G, L)$

Remark.
If $H\leq G$, it’s obvious that the inclusion map from $H$ to $G$ is a group homomorphism.

Definition. Opposite group
Let $\G$ be a group, then $(G, (x, y)\mapsto y*x)$ is called the opposite group of $\G$.

It could be easily checked that this is well-defined.

Definition. Direct product
Let $\{(G_\alpha, *_\alpha)\mid \alpha\in I\}$ be a family of groups, then $\prod_{\alpha\in I}G_\alpha$ is a group defined under

$$*:(x, y)\mapsto (\alpha \mapsto x(\alpha)*_\alpha y(\alpha)) $$

called the direct product of this family of groups.

Remark.

• One can show that given $H$ and $K$, $H\times K$ is abelian $\iff$ both $H$ and $K$ are abelian.

Definition. Generating set
Let $G$ be a group, $S\subset G$, the intersection of all subgroups which contain all elements of $S$ is called the subgroup generated by $S$, denoted as $⟨S⟩$. The elements in $S$ are called the generating set of this subgroup.

The concept of generating set is similar to basis in vector space, but not the same because groups are weaker in expressive power compared to vector spaces.

Definition. Cyclic groups
A group $G$ is called cyclic if it can be generated by a single element.

2. Quotient Group

Definition. Fiber
If $f:A\to B$. Pick $b\in B$, $f^{-1}(b):=\{a\in A\mid f(a)=b\}$ is called a fiber of $f$ over $b$.

Definition. Kernel
If $\varphi\in \Hom(G, H)$, the kernel of $\varphi$ is defined as $\{g\in G\mid \varphi(g)=e_H\}$ , denoted by $\ker \varphi$

Definition. Coset
For any $N\leq G$ and any $g\in G$, $gN:=\{gn\mid n\in N\}$ is called a left coset of $N$ in $G$. Similarly $Ng:=\{ng\mid n\in N\}$ is a right coset of $N$ in $G$. Any element of a coset is called a representative for the set.
The set of all left cosets of $G$ modulo $N$ is denoted by $G/N$

Lemma. One can construct an equivalence relation from a given subgroup
If $H\leq G$, the the relation $\sim_H\subset G\times G$ defined by

$$\sim_H:=\{(a, b)\mid a^{-1}b\in H\} $$

is a equivalence relation by definition

1. reflexivity. $\forall a\in G, a^{-1}a =e\in H$.
2. symmetry. $\forall a, b\in G. a^{-1}b\in H\implies b^{−1}a = (a^{−1}b)^{−1}\in H$.
3. transitivity. $\forall a, b, c\in G.a^{−1}b\in H\land b^{−1}c\in H\implies a^{−1}c = (a^{−1}b)(b^{−1}c)\in H$.

Lemma. Left cosets are equivalence classes, i.e.

$$H\leq G\implies G/\sim_H=G/H $$

Pf. By definition

$$\begin{aligned} \left[g\right]&=\{g'\in G\mid g^{-1}g'\in H\}\\ &=\{g'\in G\mid \exists h\in H. g^{-1}g'=h\}\\ &=\{g'\in G\mid \exists h\in H. g'=gh\}\\ &=\{gh\in G\mid h\in H\}\\ &=gH \end{aligned} $$

This lemma is telling us $g_1H=g_2H\iff g_1^{-1}g_2\in H$ if $g_1,g_2\in G$ and $H\leq G$.

Lemma. Any two left cosets are equinumerous
That is, $G$ is a group, $H\leq G$, then there exists a bijection between $g_1H$ and $g_2H$ where $g_1,g_2\in G$.
Pf. WLOG, we only need to show that there’s a bijection btw $H$ and $gH$.
Consider $f:h\mapsto gh$. This is a surjection because

$$gH:=f(H) $$

This is an injection because

$$f(h_1)=f(h_2)\iff gh_1=gh_2\implies h_1=h_2 $$

This lemma indicates that if $|G|<\infty$, all the left cosets of $H$ then have the same cardinality

Definition. Normal subgroup
If $N\leq G$, and $N$ satisfies that $\forall g\in G.gNg^{-1}\in N$ (written as $gNg^{-1}\subset N$), then $N$ is called a normal subgroup of $G$, denoted by $N\Norm G$.

Definition. Conjugate, Normalize
If $N\leq G$, the element $gng^{-1}$ is called the conjugate of $n\in N$ by $g$. The set $gNg^{-1}$ is called the conjugate of of $N$ by $g$. An element $g\in G$ is said to normalize $N$ if $gNg^{-1}=N$. In this sense, a subgroup $N$ is said to be a normal subgroup of $G$ iff all the element of $G$ normalizes $N$.

Theorem. Kernels are normal subgroups
If $H\leq G$, TTFE

1. There is a surjective homomorphism $f: G\to Q$ such that $H = \Ker(f)$
2. $H\Norm G$
   Pf. ($1\implies 2$) Let $H = \Ker(f)$ for some group homomorphism $f$, then

$$\forall h ∈ H,\forall g∈ G. f(ghg^{−1}) = f(g)f(h)f(g^{−1}) = f(g)e_Qf(g)^{-1}=e_Q $$

so $ghg^{−1}\in H$.
($2\implies 1$) Let $Q = G/H$ and $f$ be the quotient map, i.e.

$$f:g\mapsto gH $$

and define the group operation

$$\begin{aligned} \cdot: Q\times Q&\to Q \\ [a]\cdot [b]&\mapsto[ab] \end{aligned} $$

Firstly we show that $·$ is well defined. Suppose $aH = a'H$, $bH = b'H$, then $a^{−1}a'\in H$, $b^{−1} b'\in H$, and

$$(ab)^{−1}(a'b') = b^{−1}a^{−1}a'b' = (b^{−1}(a^{−1}a')b)(b^{−1}b')\in H $$

Hence $(ab)H = (a'b')H$.
Then we want to show $Q$ is a group, and $f\in \Hom(G,Q)$

• Associativity follows from the associativity of group operation on $G$
• $e_Q = e_GH$
• $(aH)^{−1} = (a^{-1})H$
  It is evident from the construction that the quotient map $G → G/\sim_H= Q$ is now a surjection and a group homomorphism, and its kernel is $H$.

Recall that in linear algebra, the kernel of a linear transformation is a subspace of the domain space. This theorem is telling exactly the same thing.

Definition. Quotient group
If $N\Norm G$, then the set of all left-cosets of $N$ (denoted by $G/N$) is a group (as we have shown in the above theorem) with the operation $*$ defined as follows

$$\forall g,h\in G. gN*hN\mapsto (gh)N $$

We call $G/N$ the quotient group of $G$ modulo $N$

Note that $G/N$ is a well-defined group only when $N\Norm G$. If we only have $N\leq G$, this set of left cosets is not necessarily a group.

3. Isomorphism Theorems

Theorem. First Isomorphism Theorem.
If $\varphi\in \Hom(G, H)$, then $\Ker(\varphi)\Norm G$ and $G/\Ker(\varphi)\Norm \varphi(G)$

Theorem. Second (Diamond) Isomorphism Theorem
Theorem. Third Isomorphism Theorem
Theorem. Fourth (Lattice) Isomorphism Theorem

Remark.

1. Any subgroup of an abelian group is a normal subgroup
2. Let $n$ be an integer, $⟨n⟩ = \{nz \mid z\in \Z\}$ is a normal subgroup of $(\Z, +)$ (because of commutativity) the quotient group $\Z/⟨n⟩$ is denoted as $\Z/n$, $\Z_n$ or $\Z/n\Z$. By definition, these quotient groups are cyclic. Specially, we use $(\Z/n\Z)^\times$ to denote all the elements in $\Z/n\Z$ that contains multiplicative inverses.

4. Semidirect Product

The concept of semidirect product is much more complicated than that of direct product. Given two groups $H$ and $K$, the semidirect product of $H$ and $K$ is more cunning because

• the semidirect product can be non-abelian even both $H$ and $K$ are abelian
• there can be multiple non-isomorphic semidirect products using the same two groups

Definition. Semidirect product
Let $H$ and $K$ be groups and let $\varphi\in\Hom(K, \Aut(H))$. Let $\cdot$ denote the left action of $K$ on $H$ determined by $\varphi$. Let $G$ be the ordered pairs $(h, k)\in H\times K$ and define the following multiplication on $G$

$$(h_1,k_1)(h_2,j_2)\mapsto (h_1k_1\cdot h_2, k_1k_2) $$

It could be proved that

1. This multiplication makes $G$ into a group of order $|G|=|H||K|$
2. The sets $\{(h,1)\mid h\in H\}$ and $\{(1,k)\mid k\in K\}$ are subgroups of $G$ and there are obvious isomorphisms by project functions

$$H\Iso\{(h,1)\mid h\in H\}\quad\quad K\Iso\{(1,k)\mid k\in K\} $$

By identifying $H$ and $K$ and their identities described in 2 we have the following properties

• $H\Norm G$
• $H\cap K=1$
• $\forall h\in H.\forall k\in K.khk^{-1}=k\cdot h=\varphi(k)(h)$

Such a $G$ is called the semidirect product of $H$ and $K$ with respect to $\varphi$ and will be denoted by

$$G=H\rtimes_{\varphi} K $$

When there’s no danger of confusion, we write

$$G=H\rtimes K $$

To distinguish, there’s a similar but different concept of groups as follows

Definition. Direct sum
A group $G$ is called the direct sum of two subgroups $H$ and $K$ if

• $H\Norm G,K\Norm G$
• $H\cap K=\{e\}$
• $G = ⟨H, K⟩$; in other words, G is generated by the subgroups $H$ and $K$.