NoteNextra-origin/content/Math4302/Math4302_L13.md

Math4302 Modern Algebra (Lecture 13)

Groups

Cosets

Last time we see that (left coset) a\sim b (to differentiate from right coset, we may denote it as a\sim_L b) by a^{-1}b\in H defines an equivalence relation.

Definition of Equivalence Class

Let a\in H, and the equivalence class containing a is defined as:

aH=\{x|a\simeq x\}=\{x|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}
$$.

#### Properties of Equivalence Class

$aH=bH$ if and only if $a\sim b$.

<details>
<summary>Proof</summary>

If $aH=bH$, then since $a\in aH, a\in bH$, then for some $h$, $a=bh$, since $b^{-1}a\in H$, so $a^{-1}b\in H$, therefore $a\simeq b$.

If $a\sim b$, then $aH\subseteq bH$, since anything in $aH$ is related to $a$, therefore it is related to $b$ so $a\in bH$.

$bH\subseteq aH$, apply the reflexive property for equivalence relation, therefore $b\in aH$.

So $aH=bH$.

</details>

If $aH\cap bH\neq \emptyset$, then $aH=bH$.

<details>
<summary>Proof</summary>
If $x\in aH\cap bH$, then $x\sim a$ and $x\sim b$, so $a\sim b$, so $aH=bH$.

</details>

$aH=H$ if and only if $a\in H$.

<details>
<summary>Proof</summary>
$aH=eH$ if and only if $a\sim e$, if and only if $a\in H$.

</details>

$aH$ is called **left coset** of $a$ in $H$.

<details>
<summary>Examples</summary>

Consider $G=S_3=\{e,\rho,\rho^2,\tau_1,\tau_2,\tau_3\}$.

where $\rho=(123),\rho^2=(132),\tau_1=(12),\tau_2=(23),\tau_3=(13)$.

$H=\{e,\rho,\rho^2\}$.

All the left coset for $H$ is $H=eH=\rho H=\rho^2H$.

\tau_1\rho=(23)=\tau_2\ \tau_1\rho^2=(13)=\tau_3\ \tau_2\rho=(31)=\tau_3\ \tau_2\rho^2=(12)=\tau_1 \tau_3\rho=(12)=\tau_1\ \tau_3\rho^2=(23)=\tau_2

\tau_1H={\tau_1,\tau_2,\tau_3}=\tau_2H=\tau_3H\

---

Consider $G=\mathbb{Z}$ with $H=5\mathbb{Z}$.

We have 5 cosets, $H,1+H,2+H,3+H,4+H$.

</details>

#### Lemma for size of cosets

Any coset of $H$ has the same cardinality as $H$.

Define $\phi:H\to aH$ by $\phi(h)=ah$.

$\phi$ is an bijection, if $ah=ah'\implies h=h'$, it is onto by definition of $aH$.

#### Corollary: Lagrange's Theorem

If $G$ is a finite group, and $H\leq G$, then $|H|\big\vert |G|$. (size of $H$ divides size of $G$)

<details>
<summary>Proof</summary>

Suppose $H$ has $r$ distinct cosets, then $|G|=r|H|$, so $|H|$ divides $|G|$.

</details>

#### Corollary for Lagrange's Theorem

If $|G|=p$, where $p$ is a prime number, then $G$ is cyclic.

<details>
<summary>Proof</summary>

Prick $e\neq a\in G$, let $H=\langle a\rangle \leq G$, then $|H|$ divides $|G|$, since $p$ is prime, then $|H|=|G|$, so $G=\langle a \rangle$.

</details>

If $G$ is finite and $a\in G$, then $\operatorname{ord}(a)\big\vert|G|$.

<details>
<summary>Proof</summary>

Since $\operatorname{ord}(a)=|\langle a\rangle|$, and $\langle a\rangle $ is a subgroup, so $\operatorname{ord}(a)\big\vert|G|$.

</details>

#### Definition of index

Suppose $H\leq G$, the number of distinct left cosets of $H$ is called the index of $H$ in $G$. Notation is $(G:H)$.

#### Definition of right coset

Suppose $H\leq G$, define the equivalence relation by $a\sim 'b$ (or $a\sim_R b$ in some textbook) if $a b^{-1}\in H$. (note the in left coset, we use $a^{-1}b \in H$, or equivalently $b^{-1}a \in H$, these are different equivalence relations)

The equivalent class is defined

Ha={x\in G|x\sim'a}={x\in G|xa^{-1}\in H}={x|x=ha\text{ for some }h\in H}

Some properties are the same as the left coset

- $Ha=H\iff a\in H$
- $Ha=Hb$ if and only if $a\sim'b\iff a b^{-1}\in H$.
- $Ha\cap Hb\neq \emptyset\iff Ha=Hb$.

Some exercises: Find all the left and right cosets of $G=S_3$, there should be 2 left cosets and 2 right cosets (giving different partition of $G$).