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 $a\sim b$ .

Proof

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 $aH=bH$ .

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

Proof

If $x\in aH\cap bH$, then $x\sim a$ and $x\sim b$, so $a\sim b$, so $aH=bH$.

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

Proof

$aH=eH$ if and only if $a\sim e$, if and only if $a\in H$.

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

Examples

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, $H,1+H,2+H,3+H,4+H$ .

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 $G$ )

Proof

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

Corollary for Lagrange’s Theorem

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

Proof

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 $G=\langle a \rangle$ .

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

Proof

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

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 $G$ ).