MATH 415 Lecture 11

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Factor Groups

Theorem 14.1. Let $\phi :G\to G'$ be a homomorphism and $H$ be the kernel of $\phi$ . We denote $G/H$ as the set of left cosets (identical to right cosets since $H$ is the kernel).

We call $\left\langle G/H,*\right\rangle$ the factor group obtained from $G$ and the homomorphism $\phi$ .

we define $*$ as the operation $(a\,H)*(b\,H)=(ab)\,H$

The function $\mu :G/H\to \phi [G]$ is an isomorphism on the image of $\phi$ .

Similarly, we define $H\setminus G=\left\{H\,g~\mid ~g\in G\right\}$ , and note that $G/H\neq H\setminus G$ in general.

Theorem 14.4

Recall that a subgroup $H\trianglelefteq G$ is normal if its left and right cosets coincide.

Let $H\leq G$ . Then left coset multiplication is well-defined by the equation $(a\,H)*(b\,H)=(ab)\,H$ if and only if $H$ is a normal subgroup of $G$ .

Proof. Suppose that $*$ does not give a well-defined binary operation; so for $a\in G$ , is it true that $aH=Ha$ ? Let $x\in aH$ , $a\in aH$ , and $a^{-1}\in a^{-1}H$ . By definition $x$ and $a$ are from the same coset.

${\begin{aligned}(xH)(a^{-1}H)&=(xa^{-1})H\\(aH)(a^{-1}H)&=eH=H\\(xH)(a^{-1}H)&=H\\xa^{-1}&=h\in H\\x&=ha\end{aligned}}$

Therefore, $x\in Ha$ , when we have assumed it to be a member of a left coset. Therefore, $aH\subseteq Ha$ and $Ha\subseteq aH$ imply that $aH=Ha$ and thus $H$ is normal.

So if $H$ is normal in $G$ , let's take a look at the operation $(aH)(bH)=(ab)H$ . Is it true that if $a_{1}\in aH$ and $b_{1}\in bH$ , then $(a_{1}H)(b_{1}H)=(a_{1}b_{1})H=(ab)H$ ?

Using our definition of $a_{1}$ and $b_{1}$ , we arrive at the following: $a_{1}\,b_{1}=a\,h_{1}\,b\,h_{2}$ . If we look at $h_{1}\,b$ , we deduce that $h_{1}\,b$ must be equivalent to $b\,h_{3}$ for some $h_{3}$ since $H$ is normal and its cosets coincide. Therefore $a\,h_{1}\,b\,h_{2}=a\,b\,h_{3}\,h_{2}$ . Since $H$ is closed on $*$ , we can represent $h_{3}\,h_{2}$ as $h'\in H$ , therefore $(a_{1}\,b_{1})H$ is equivalent to $(a\,b)H$ .

quod erat demonstrandum

Corollary. If $H\trianglelefteq G$ , then the cosets of $H$ form a group $G/H$ under the binary operation $(aH)(bH)=(ab)H$ .

Associativity: $((aH)(bH))(cH)=((ab)H)(cH)=((ab)c)H=(a(bc))H=(aH)((bc)H)=(aH)((bH)(cH))$
Identity: $e=eH=H$ , which is the ID element in $G/H$
Inverse: $(aH)^{-1}=a^{-1}H$ , so $(aH)(a^{-1}H)=(a\,a^{-1})H=eH=H=id$

Examples

$G=\mathbb {Z}$ , and $n\mathbb {Z} =H\trianglelefteq G$

What is the quotient group $G/H$ ?

$G/H=\mathbb {Z} /n\mathbb {Z}$ . This group is exactly isomorphic to $\mathbb {Z} _{n}$ on addition mod $n$ .

The Fundamental Homomorphism Theorem

First, we need Theorem 14.9. Let $H\trianglelefteq G$ . Then $\gamma :G\to G/H$ given by $\gamma (x)=xH$ is a homomorphism with the kernel $H$ .

Proof. For $x,y\in G$ , we need to check that $\gamma (x\,y)=\gamma (x)\,\gamma (y)$ . By definition $\gamma (x\,y)=(x\,y)H=(xH)(yH)$ . Observe that the first factor is $\gamma (x)$ and the second is $\gamma (y)$ . Therefore it is a homomorphism.

What about kernel? By definition, the kernel is $\left\{x\in G~\mid ~\gamma (x)=e_{G/H}\right\}$ . Now $\gamma (x)=xH$ . Since $x\in H$ , we get $xH=eH$ , so the kernel of $\gamma$ is $H$ . Furthermore, we can say that $\gamma$ is onto.

quod erat demonstrandum

From before, we have $\mu (gH)\simeq \phi (g)$

Now we can state that $\phi =\mu \circ \gamma$ since $\phi :G\to G'$ (in particular, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi[G] \subseteq G'} ), $\mu :G/H\to \phi [G]$ , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma : G \to G/H} .

Theorem 14.11. [The Fundamental Homomorphism Theorem]. Let

\phi :G\to G'

be a group homomorphism with kernel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . Then

\phi [G]

is a group, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu : G/H \to \phi[G]} given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(gH) = \phi(g)} is an isomorphism. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma : G \to G/H} is the homomorphism given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(g) = gH} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(g) = (\mu \circ \gamma)(g)} for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \mathbb{Z}_4 \times \mathbb{Z}_2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \left\{ 0 \right\} \times \mathbb{Z} = \left\{ \left( 0,x \right) ~\mid~ x \in \mathbb{Z}_2 \right\}} .

Note Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| H \right| = 2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| G \right| = 8} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / H \simeq \mathbb{Z}_4 / \left\{ 0 \right\} \times \mathbb{Z}_2 / \mathbb{Z}_2} for a very specific reason: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = K \times L} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = K_1 \times L_1} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1 \le K} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_1 \le L} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H = K / K_1 \times L / L_1} .

Back to the example, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_4 / \left\{ 0 \right\} = \mathbb{Z}_4} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2 / \mathbb{Z}_2 = \left\{ 0 \right\}} because

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ e \right\} \trianglelefteq G} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \trianglelefteq G}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / \left\{ e \right\} = G}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / G = \left\{ e \right\}}

And now Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2 \times \left\{ 0 \right\} = \mathbb{Z}_2} because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \times \left\{ e \right\} = G} .

Theorem 14.13

The following are three equivalent conditions for a subgroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} of a group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} to be a normal subgroup of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \, h \, g^{-1} \in H} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \in H} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \, H \, g^{-1} = H} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH = Hg} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} .

Note: The transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to g \, h \, g^{-1}} is called the conjugation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} .

Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \trianglelefteq G} , so 3 holds. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \in H} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \, h \, g^{-1} = h_1} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \, h = h_1 \, g} which holds by the assumption.

Automorphisms

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Aut}(G)} be the group of automorphisms, or isomorphisms of a group with itself, over the composition operation.

It is closed over composition since if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : G \to G} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi : G \to G} are isomorphisms, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \circ \psi : G \to G} will be an isomorphism as well.
We already know that composition is associative
The identity element will be the identity isomorphism (mapping elements in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} to themselves)