Groups

In this note, I will briefly introduce groups, and some of their properties. Group Action would be talked in another seperate blog.
I took Abstract Algebra in UW-Madison(MATH541) with professor Chenxi Wu. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

Groups, Subgroups

Let's first introduct some concepts and facts about group.

Definition 1. [Group, Abelian group, Trivial group, Permutation group]

• A group $(G, *)$ is a combination of a set $G$ and a map $*: G \times G \to G$, also called group operation. The group operation are require to have:

  1. ("associativity") For any $a, b, c\in G$, $*(*(a, b), c)=*(a, *(b, c))$;

  2. ("identity") There is an element $e\in G$, such that for any $a\in G$, $*(e, a)=*(a, e)=a$. This element is also written as $id_G$;

  3. ("inverse") For any $a\in G$, there is a $b\in G$, such that $*(a, b)=*(b, a)=e$.

  Group operation $*(a, b)$ can also be written as $a * b$ or $ab$.

• An abelian group has one more property:

  1. ("communitation") For any $a, b\in G$, $*(a, b)=*(b, a)$.

There are some special groups.

Definition 2. [Opposite Group, Trivial Group, Permutation Group]

• An opposite group is $(G, *')$ with $*'(a, b) = *(b, a)$;

• A trivial group is $(\{e\}, (e, e)\mapsto e)$;

• The set of bijections from set $A$ to $A$ itself forms a group under operation $\circ$, which is called permutation group, denoted as $S_{A}$. When $A = {1, 2, 3, ..., n}$, the permutation group can also be represented as $S_{n}$;

Groups are fundamentally set, which contains subsets. With the same operation as the original multiplication, the subsets can also form groups as they are "closed". Those subsets are called "subgroups".

Definition 3. [Subgroup]

Let $(G, *)$ be a group. A subset $H\subseteq G$ is called a subgroup, denoted as $H\leq G$, if

1. $e_{G} \in H$.

2. $\forall \,a, b\in H$, $a*b\in H$.

3. $\forall \,a\in H$, $a^{-1}\in H$.

For any subgroup $H \leq G$, it's clear that we can define a new operation $*|_{H\times H}$, which is almost the same as the original $*$ but limit $*(a, b)$ to $a, b\in H$.

Homomorphism, Isomorphism, Automorphism

Till now, we are only talking about relationships in groups. However, groups are not totally isolated: we can connect them as how we use functions to connect sets. However, these functions are special as groups are not simply sets. We call these functions "homomorphism".

Definition 4. [Homomorphism, Isomorphism, Endomorphism, Automorphism]

• A homomorphism is a function $f: G\rightarrow H$: $(G, *)$ and $(H, *')$ be two groups, and for any $a, b\in G$, $f(a*b)=f(a)*'f(b)$.

  • $Hom(G, H)$ is the set of homomorphisms between two groups $G$ and $H$.

• An isomorphism is a bijective homomorphism. Specificly, if $f: G \rightarrow H$ is an isomorphism, $f^{-1}: H \rightarrow G$ is also a isomorphism. We say $G$ is isomorphic to $H$ if there is an isomorphism between $G$ and $H$, with notation $G \cong H$.

• An endomorphism is a homomorphism from $G$ to itself.

• An automorphism is a isomorphism between a group and itself. Also, the set of all automorphisms of a group $G$ forms a group, called automorphism group, written as $Aut(G)$.

Isomorphic and automorphism group are two concepts that we will use a lot in the following instructions and many problems in Abstract Algrebra. The reason is, $G \cong H$ means the two groups are very "similar". You can shape every element in $G$ to an unique element in $H$, and you can even apply multiplications on them, and get the corresponding product!

Till now, we have introduced almost every concept that we will be using in this blog. However, this is far not enough to end here - we only have some abstract, useless, non-sense definition. Why we need to define group? Why we need to define (homo)(iso)(auto)morphisms? It's impossible to answer, unless we can find some more "fun facts" and applications about them.

Isomorphism Theorem

First let's define two concepts kernal and image of a homomorphism.

Definition 5. [Kernal, Image]

Let $f: G \rightarrow H$ be a group homomorphism, we have two special group:

• $\{g \in G: f(g) = e_H\}$, or $f^{-1}(e_H)$ is called the kernal of $f$. It can also be written as $ker(f)$. Specificly, $ker(f) \leq G$;
• $f(G)$ is called the image of $f$, which can be written as $im(f)$. Also, $im(f) \leq H$.

Theorem 6. Let $f: G \rightarrow H$ be a group homomorphism, we have:

• $f$ can be written as the composition of $f_1: G \rightarrow f(G)$ with $f_1(g) = f(g)$ and $f_2: f(G) \rightarrow H$ with $f_2(h) = h$. Clearly, $f_1$ is a surjection and $f_2$ is an injection;
• $f(a) = f(b)$ iff $a^{-1}b \in ker(f)$. This is clear as $f(ab^{-1}) = f(a)f(b^{-1}) = f(b)f(b^{-1}) = e$;

Theorem 6. Let $f: G \rightarrow H$ be a group homomorphism and $g: G \rightarrow Q$ be a surjective group homomorphism, we have:

1. There is a unique group homomorphism $h: Q \rightarrow H$ such that $f = h \circ g$ if and only if $ker(g) \subseteq ker(f)$;
2. $h$ is surjective iff $f$ is surjective;
3. $h$ is injective iff $ker(g) = ker(f)$;
4. If the two conditions are both satisfied, we have $h$ is a isomorphism.

Definition 7. [Left coset]

Let $G$ be a group, $H \leq G$ be a subgroup, then we have $\sim_H = \{(a, b) \in G \times G \; : \; a^{-1}b \in H\}$ is a valid equivalence relation. The equivalence classes can be represented as $[g]$ or $gH$, and we call them left cosets.

Definition 8. [Normal subgroup, Quotient group]

Let $H \leq G$ be a subgroup, then the following are equivalent:

1. $\forall g \in G, h \in H$, $ghg^{-1} \in H$;
2. There is a surjective homomorphism $f: G \rightarrow Q$ such that $H = ker(f)$.

For any subgroups that satiesfy the condition, we call then normal subgroup, and the left cosets of $H$ forms a group under $[g_1] \times [g_2] = [g_1g_2]$. We remark the group as $G/H$, called quotient group.

Theorem 9. [Isomorphism Theorem] Let $f:G \rightarrow H$ be a group homomorphism, then $f(G) \cong G / ker(f)$. If we have $ker(f)=\{e_G\}$, $G \cong f(G)$.

This theorem is powerful as for any $f:G \rightarrow H$, we can find a subgroup of $H$ isomorphic to quotient group of $G$.

Semidirect Product

Theorem 7. Let $G, H, N$ be three groups, then the following are equivalent:

1. $G \cong H \times N$;
2. There is an injective homomorphism $s: H \rightarrow G$ and a surjective homomorphism $p: G \rightarrow H$, such that $s(H) \trianglelefteq G$, $ker(p) \cong N$, and $p \circ s = id_H$.
3. There exists two normal subgroups $H' \trianglelefteq G, N' \trianglelefteq G$, such that $H \cong H'$, $N \cong N'$, with $H' \cap N' = \{e_{G}\}$, and for any $g \in G$, $\exists h \in H', n \in N'$, such that $g = hn$;

This theorem can be proved by $(1) \Rightarrow (2), (2) \Rightarrow (3), (3) \Rightarrow (1)$.

Why this is called "semidirect product"? As we see in the previous examples, $G = H \times N$ is obviously a group under the group operations in $H$ and $N$. However, for some group $G$, it can also be written as the product of two groups $H$ and $N$, with each element uniquely represented by $hn$. Semidirect product is obviously not the same as direct product, but it could be represented

Definition 8. [Section, Semidirect product]

The injective homomorphism $s$ in the above theorem is called a section, and $G$ is called the semidirect product between $H$ and $N$, written as $G \cong N \rtimes H$.

The following part requires some basic knowledge of group actions. Please read the parts about "permutation representation". Please read Group Action first.