# Math4302 Modern Algebra (Lecture 18)

## Groups

### Factor group

Suppose $G$ is a group, and $H\trianglelefteq G$, then $G/H$ is a group.

Recall from last lecture, if $\phi:G\to G'$ is a homomorphism, then $G/\ker(\phi)\simeq \phi(G)\leq G'$.

<details>
<summary>Example (continue from last lecture)</summary>

$\mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle\simeq \mathbb{Z}$

Take $\phi(a,b)=a-b$, this is a surjective homomorphism from $\mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle$ to $\mathbb{Z}$

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$\mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle\simeq \mathbb{Z}$

where $\langle (2,1)\rangle=\{(2b,b)|b\in \mathbb{Z}\}$

Take $\phi(a,b)=a-2b$, this is a surjective homomorphism from $\mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle$ to $\mathbb{Z}$

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$\mathbb{Z}\times\mathbb{Z}/\langle (2,2)\rangle$

This should also be a finitely generated abelian group. ($\mathbb{Z}_2\times \mathbb{Z}$ actually)

Take $\phi(a,b)=(a\mod 2,a-b)$

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More generally, for $\mathbb{Z}\times \mathbb{Z}/\langle (a,b)\rangle$.

This should be $\mathbb{Z}\times \mathbb{Z}_{\operatorname{gcd}(a,b)}$

Try to do section by gcd.

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> - If $G$ is abelian, $N\leq G$, then $G/N$ is abelian.
> - If $G$ is finitely generated and $N\trianglelefteq G$, then $G/N$ is finitely generated.

#### Definition of simple group

$G$ is simple if $G$ has no proper ($H\neq G,\{e\}$), normal subgroup.

> [!TIP]
> 
> In general $S_n$ is not simple, consider the normal subgroup $A_n$.

<details>
<summary>Example of some natural normal subgroups</summary>

If $\phi:G\to G'$ is a homomorphism, then $\ker(\phi)\trianglelefteq G$.

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The **center** of $G$: $Z(G)=\{a\in G|ag=ga\text{ for all }g\in G\}$

$Z(G)\trianglelefteq G$.

- $e\in Z(G)$.
- $a,b\in Z(G)\implies abg=gab\implies ab\in Z(G)$.
- $a\in Z(G)\implies ag=ga\implies a^{-1}\in Z(G)$.
- If $g\in G, h\in Z(G)$, then $ghg^{-1}\in Z(G)$ since $ghg^{-1}=gg^{-1}h=h$.

$Z(S_3)=\{e\}$, all the transpositions are not commutative, so $Z(S_3)=\{e\}$.

$Z(GL_n(\mathbb{R}))$? continue on friday.

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