MATH 415 Lecture 10

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Homorphism

Let $\left\langle G,\cdot \right\rangle$ and $\left\langle G',*\right\rangle$ be groups. The function $\phi :G\to G'$ is a homomorphism if and only if for all $x,y\in G$

$\phi (x\cdot y)=\phi (x)*\phi (y)$

Example: Integration

Let $F$ represent the group of all functions over addition.

$\phi :F\to \mathbb {R}$ where $\phi (f)=\int _{a}^{b}f(x)\,\mathrm {d} x$ is a homomorphism:

$\int _{a}^{b}(f+g)(x)\,\mathrm {d} x=\int _{a}^{b}f(x)\,\mathrm {d} x+\int _{a}^{b}g(x)\,\mathrm {d} x$

Images and Ranges

Let $\phi :X\to Y$ and $A\subseteq X$.

• $\phi \left[A\right]=\left\{\phi (a)~\mid ~a\in A\right\}$ is the image of $\phi$ over $A$
• $\phi \left[X\right]$ is the range of $\phi$.

It is also possible to find inverse images:

$\phi ^{-1}\left[B\right]=\left\{x\in X~\mid ~\phi (x)\in B\right\}$

Theorem 13.12

$\phi :G\to G'$ is a homomorphism

1. if $e$ is the identity element in $G$, then �$\phi (e)=e'$ is the identity in $G'$.
2. if $a\in G$, then $\phi (a^{-1})=\phi (a)^{-1}$.
3. if $H$ is a subgroup of $G$, then $\phi \left[H\right]$ is a subgroup in $G'$.
4. if $K'$ is a subgroup of $G'$, then $\phi ^{-1}\left[K'\right]$ is a subgroup in $G$.

Proof.

1. $\phi (a)=\phi (a\,e)=\phi (a)\,\phi (e)=\phi (a)\,e'$. By transitivity, $\phi (a)=\phi (a)\,e'$, so $\phi (e)=e'$.
2. Since $e'=\phi (e)=\phi (a\,a^{-1})$, we can say $\phi (a\,a^{-1})=\phi (a)\,\phi (a^{-1}=e'$. Therefore $\phi (a^{-1})=\phi (a)^{-1}$.
3. Let $\phi (a),\phi (b)\in \phi \left[H\right]$ for $a,b\in G$. Since $G$ is a group, then $a\,b\in G$ and thus $\phi (a\,b)=\phi (a)\,\phi (b)\in \phi \left[H\right]$.Furthermore, if $\phi (a)\in \phi \left[H\right]$, then $\phi (a^{-1})=\phi (a)^{-1}\in \phi \left[H\right]$.

quod erat demonstrandum

Kernel

Let $\phi :G\to G'$ be a homomorphism. $\left\{e'\right\}$ is the trivial subgroup of $G'$.

We define $\phi ^{-1}\left[\left\{e'\right\}\right]=\left\{x\in G~\mid ~\phi (x)=e'\right\}$ to be the kernel of $\phi$.

The kernel is a subgroup in $G$ that maps all of its members to $e'$ in $G'$. Thus the kernel is collapsing down to $e'$.

Example

Let $\phi :\mathbb {R} ^{n}\to \mathbb {R} ^{m}$, where $\phi ({\vec {x}})=A\,{\vec {x}}$, where $A$ is a $m\times n$ matrix.

The kernel of $\phi$ is the null space of $A$

Theorem 13.15

Let $\phi :G\to G'$ be a homomorphism, $H=\mathrm {kernel} (\phi )$, and let $a\in G$. Then the set $\phi ^{-1}\left[\left\{\phi (a)\right\}\right]=\left\{x\in G~\mid ~\phi (x)=\phi (a)\right\}$ is the left coset $a\,H$ and at the same time is the right coset $H\,a$ (thus implying $a\,H=H\,a$)

Proof. Consider 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 x \in a\,H} , thus 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 x = a\,h} for 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} . 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 \phi} is a homomorphism, 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 \phi(x) = \phi(a\,h) = \phi(a) \, \phi(h) = \phi(a) \cdot e' = \phi(a)} .

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 x \in A} , 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(x) = \phi(a)} , 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(x) \, \phi(a)^{-1} = e' = \phi(x) \, \phi(a^{-1})} . Therefore 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 (x \, a^{-1} = h) \in (H = \mathrm{kernel}(H))} implies 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 x = h\,a} , 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 x \in H\,a} . a similar strategy shows 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 x \in a\,H} . Thus the right and left cosets are identical.

quod erat demonstrandum

Example

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 D} be the group of differentiable mappings from 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{R} \to \mathbb{R}} over addition.

Therefore 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 f'} is well-defined. Consider 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 : D \to D} with 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(f) = f'} .

Observe 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 \phi} is a homomorphism because $\phi (f+g)=\phi (f)+\phi (g)=f'+g'$.

The kernel of $\phi$ is the set of all functions whose derivative is 0, thus it is the set of constant functions.

Corollary

Corollary. A group homomorphism $\phi :G\to G'$ is injective if and only if its kernel is $\left\{e\right\}$.

If $\left|\mathrm {kernel} (\phi )\right|>1$, then $\phi$ is not one-to-one since $x,y\in \mathrm {kernel} (\phi )$ for $x\neq y$. Alternatively, if $\mathrm {kernel} (\phi )=\left\{e\right\}$, then assuming $\phi (a)=\phi (x)=e$, we arrive at the contradiction that the kernel must have at least two elements.

Relation to Isomorphism

Recall that if $\phi :G\to G'$ is an isomorphism, then

1. $\phi$ is a homomorphism
2. $\phi$ is injective (hence its kernel is $\left\{e\right\}$)
3. $\phi$ is surjective

Normal Subgroups

A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide. That is, if $g\,H=H\,g$ for all $g\in G$.

Notation: $H\trianglelefteq G$

Corollary. The kernel of a homomorphism $\phi$ is a normal subgroup of $G$.

Factor Groups

Given a group $G$ partitioned into cosets, we want to collapse all cosets $g\,H$ (for $g\in G$) into individual elements in another group $G'$. In general, this is not possible, but it is possible if $H$ is a normal subgroup.

Theorem 14.1

Let $\phi :G'$ be a group homomorphism 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 = \mathrm{kernel}(\phi)} . Then the cosents 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} form a group (called a factor group; denoted 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} ), where 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 (a\,H) \,(b\,H) = (a\,b) \, H} . The identity element is $e\,H=H=\mathrm {kernel} (\phi )$.

Furthermore, the map 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\left[ G \right]} defined 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(a\,H) = \phi(a)} is an isomorphism.

Both coset multiplication 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} are well-defined independent of the choices 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 a} and $b$ from the cosets.

Note: It's important to note that each coset contains multiple elements, so choosing any representative still refers to the same coset.

Proof. By definition, 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 (a\,H)(b\,H) = (a\,b)\,H} . What about 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 (a_1 \, H)(b_1 \, H)} ? 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 a_1 \,H = a\,H} , then there exists an 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_1 \in H} such that $a_{1}=a\,h_{1}$. Similarly, 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 b_1 \in b\,H} , 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 b_1 = b \, h_2} for 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_2 \in H} . Thus 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 a_1 \, b_1 = a \, h_1 \, b \, h_2} Since $h_{1}\,b\in H\,b$ 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} is a normal subgroup, we can say 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 \, b = b \, H} , 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 h_1 \, b \in b \, H} implies 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_1 = b \, h_3} for some $h_{3}\in b\,H$. Thus 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 a_1 \, b_1 = a \, b \, h_3 \, h_2} . 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} is closed, so 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 h_3 \, h_2 = h' \in H} . Thus 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 (a_1 \, b_1)\,H = (a\,b) \, H} .

…

quod erat demonstrandum