MTH3003 Lecture 9

Following our exploration of normal subgroups, we now look at how to use them to “divide” groups and construct entirely new ones. This naturally leads us to formalise structure-preserving maps between groups, giving us the tools to prove when two groups are structurally identical.

Quotient Groups

Suppose we have a subgroup $H\le G$. We want to construct a new group by “dividing” $G$ by $H$, forming a set of cosets $G/H$. However, this set is not always a group! It only forms a valid group when $H$ is a normal subgroup ($H⊴G$). When this condition is met, the resulting structure is called a quotient group.

Definition: Quotient Group

Suppose $N⊴G$. The quotient group $G/N$ is the set of cosets of $N$ in $G$ with the operation defined for all $aN,bN\in G/N$ as: $\boxed{aN\ast bN=(ab)N}$

Notice that even without this definition, $(aN)(bN)=a(Nb)N=a(bN)N=abN$ holds strictly because $N$ is normal.

To show that $G/N$ is a group, we check the axioms for $aN,bN,cN\in G/N$:

• Closure: $(ab)N$ is clearly a coset of $N$ in $G$, so it is in $G/N$.
• Identity: We define the identity $e_{G/N}$ to be the coset $e_{G}N=N$. Thus, $\boxed{e_{G/N}=N}$.
• Inverses: The inverse of $aN$ is $\boxed{(aN{)}^{-1}=a^{-1}N}$, which we can verify:

$$(a^{-1}N)\ast (aN)=(a^{-1}a)N=e_{G}N=N$$

• Associativity: Inherited directly from $G$:

$$(aN)\ast ((bN)\ast (cN))=(aN)\ast ((bc)N)=(a(bc))N=((ab)c)N=((aN)\ast (bN))\ast (cN)$$

The Quotient Group $(\mathbb{Z},+)/3\mathbb{Z}$

Since $(\mathbb{Z},+)$ is abelian, $3\mathbb{Z}$ is a normal subgroup, and the cosets are: $(\mathbb{Z},+)/3\mathbb{Z}=\{0+3\mathbb{Z},1+3\mathbb{Z},2+3\mathbb{Z}\}$ This behaves exactly like ${\mathbb{Z}}_3$ under addition modulo 3, giving $\boxed{(\mathbb{Z},+)/3\mathbb{Z}\cong {\mathbb{Z}}_3}$.

The Quotient Group $S_3/C_3$

For $C_3=\{e,(123),(132)\}⊴S_3$, there are only two cosets since $|S_3|/|C_3|=2$. $S_3/C_3=\{e{C}_3,(12)C_3\}$ Multiplying them shows $(12)C_3\ast (12)C_3=e{C}_3$, so this group behaves exactly like $C_2$.

Homomorphisms

Definition: Homomorphism

A homomorphism is a map $\theta :G\to H$ between two groups that preserves the group operation: $\boxed{\theta (g_1{g}_2)=\theta (g_1)\theta (g_2)}$ (where the operation on the left is in $G$, and on the right is in $H$).

Modular Arithmetic Homomorphism

The map $\varphi :\mathbb{Z}\to {\mathbb{Z}}_n$ defined by $\varphi (m)=[m{]}_n$ is a homomorphism. This holds because $\varphi (m_1+m_2)=[m_1+m_2{]}_n=[m_1{]}_n\oplus [m_2{]}_n=\varphi (m_1)\oplus \varphi (m_2)$.

A map $\pi :S_3\to C_3$ that maps all 2-cycles to $e$ and fixes elements of $C_3$ fails to be a homomorphism, because $\pi ((13))\pi ((12))=ee=e$, but $\pi ((13)(12))=\pi ((123))=(123)$.

Kernel and Image

To understand a homomorphism, we look at what it sends to the identity, and what it covers in the target group.

Definition: Kernel and Image

The kernel $\text{Ker}(\theta )=\{g\in G:\theta (g)=e_H\}$ is the set of elements mapping to $e_H$. The image $\text{Im}(\theta )=\{\theta (g):g\in G\}$ is the set of all output elements in $H$.

For our earlier example $\varphi :\mathbb{Z}\to {\mathbb{Z}}_n$, the kernel is $n\mathbb{Z}$ (multiples of $n$), and the image is all of ${\mathbb{Z}}_n$ (so $\varphi$ is onto).

There are several fundamental properties that hold for any homomorphism $\theta :G\to H$:

1. $\boxed{\theta (e_G)=e_H}$
2. $\theta (g^m)=(\theta (g){)}^m$ for all $g\in G,m\in \mathbb{Z}$
3. $\boxed{\text{Im}(\theta )\le H}$ (the image is a subgroup)
4. $\boxed{\text{Ker}(\theta )⊴G}$ (the kernel is a normal subgroup)

Subgroup Status

Notice that while the kernel is always a normal subgroup of $G$, the image is generally just a subgroup of $H$ (not necessarily normal).

When a homomorphism is bijective (both one-to-one and onto), we call it an isomorphism, denoted $G\cong H$. This means the groups are structurally identical.

Homomorphism from $S_3$ to $S_2$

Define $\varphi :S_3\to S_2$ where $\varphi$ sends 3-cycles to $e$, and all 2-cycles to $(12)$. Checking cases confirms this preserves the group operation. Here, $\text{Ker}(\varphi )=\{e,(123),(132)\}=C_3$ and $\text{Im}(\varphi )=S_2$.

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Pre-Lecture Notes from University Notes

• Quotient Groups: For $N⊴G$, $G/N$ is the group of cosets with operation $aN\ast bN=(ab)N$.
• The identity is $N$, and inverses are $(aN{)}^{-1}=a^{-1}N$.
• Examples include $(\mathbb{Z},+)/3\mathbb{Z}\cong {\mathbb{Z}}_3$ and $S_3/C_3\cong C_2$.
• Homomorphisms: A map $\theta :G\to H$ satisfying $\theta (g_1{g}_2)=\theta (g_1)\theta (g_2)$.
• $\text{Ker}(\theta )=\{g\in G:\theta (g)=e_H\}⊴G$.
• $\text{Im}(\theta )=\{\theta (g):g\in G\}\le H$.
• Homomorphisms map identities to identities ($\theta (e_G)=e_H$) and preserve powers ($\theta (g^m)=(\theta (g){)}^m$).
• A bijective homomorphism is an isomorphism ($G\cong H$).
• Next lecture preview: Detailed exploration of isomorphisms and their formal properties.