Homomorphisms, Normal Subgroup & Conjugation

Homomorphisms

Group Homomorphism

Let $G$ and $H$ be groups. A homomorphism $\phi :G\to H$ is a map that satisfies $\phi (ab)=\phi (a)\phi (b)$ for all $a,b\in G$.

Proposition

If $\phi :G\to H$ is a group homomorphism, then the identity of $G$ is mapped to the identity of $H$.

Kernel & Image

Let $\phi :G\to H$ be a group homomorphism. The kernel and the image of $\phi$ are defined by $\begin{array}{rl}\ker (\phi )& =\{g\in G\mid \phi (g)=e_H\}\subset G,\\ \mathrm{im}(\phi )& =\{h\in H\mid \exists g\in G,\phi (g)=h\}\subset H.\end{array}$

Proposition

A homomorphism $\phi :G\to H$ is injective if and only if $\ker (\phi )=\{e_G\}$.

Proof If $\phi$ is injective then clearly $\ker (\phi )=e$. Conversely, suppose that $\ker (\phi )=\{e_G\}$ and let $a,b\in G$ with $\phi (a)=\phi (b)$. Then $\phi (a^{-1}b)=\phi (a{)}^{-1}\phi (b)=e_H$. Hence $a^{-1}b=e_G$ and therefore $a=b$. $\square$

Remark

ker($\phi$) is normal subgroup of $G$ and im($\phi$) is subgroup of $G$.

Normal Subgroup

Conjugate

Let $G$ be a group. For any $a,b\in G$, we say $a$ and $b$ and conjugate if there exists $g\in G$ such that $a=gb{g}^{-1}$.

Normal Subgroup

Let $G$ be a group. A subgroup $N\le G$ is called normal if $gN{g}^{-1}\subset N$ for every $g\in G$. We denote a normal subgroup by $N◃G$.

e.g. Let $\phi :G\to H$ be a homomorphism, then $\ker (\phi )◃G$. Fix some $x\in \ker (\phi )$, then for all $g\in G$ we have $\phi (gx{g}^{-1})=\phi (g)\phi (x)\phi (g^{-1})=\phi (g)\phi (g{)}^{-1}=e$.

Proposition

The followings are true:

1. $N\le G$ is normal iff the right coset $Ng$ coincides with the left coset $gN$ for every $g\in G$.
2. We required in the definition that $gN{g}^{-1}\subset N$ , but it follows that $gN{g}^{-1}=N$.
3. Every subgroup of an abelian group is normal.

Simple Group

A group is simple if there’s no non-trivial normal subgroups.

Normaliser

Let $G$ be a group and $H\le G$. The normaliser is the subgroup of $G$ that $N_G(H)=\{g\in G\mid gH{g}^{-1}=H\}\supseteq H$

Proposition

$N_G(H)◃G$.

Isomorphisms

Group Isomorphism

A group homomorphism $\phi :G\to H$ is called isomorphism if it has an inverse, that is a homomorphism $\psi :H\to G$ such that $\psi \circ \phi =e_G$ and $\phi \circ \psi =e_H$.

Proposition

A homomorphism $\phi :G\to H$ is an isomorphism if and only if it is bijective.

Proposition

Any two cyclic group of the same order are isomorphic.

Lambda-Map

For every $g\in G$ define a map by left multiplication with $g$: ${\lambda }_g:G\to G,x\mapsto gx$

Proposition

• ${\lambda }_g\in \text{Sym}(G)$.
• ${\lambda }_{g^{-1}}$ is the inverse of ${\lambda }_g$.

Cayley’s Theorem

The map $\lambda :G\to \text{Sym}(G)$ given by $g\mapsto {\lambda }_g$ is an injective group homomorphism.

Proof Clearly $\lambda (x)\circ \lambda (y)=\lambda (xy)$. Set $\lambda (g)={\lambda }_g=\text{id}$ where $\text{id}$ is the identity map from $G$ to $G$. Then for all $x\in G$, we have $gx=x$. It follows that $g=1$. Thus $\lambda$ is injective. $\square$

Corollary

Every finite group is isomorphic to a subgroup of a symmetric group $S_n$.

Proof For any finite group $G$, the $\lambda$ map forms an isomorphism, thus $G\cong S_n$ for some $n$.$\square$

Automorphism

An automorphism is an isomorphism from a group $G$ to $G$. Denote $\text{Aut}(G)$ as the set of all automorphisms on $G$.

Conjugation

Conjugation

Let $G$ be a group and $g\in G$. The map${\gamma }_g:G\to G,a\mapsto ga{g}^{-1}$is called conjugation by $g$. We say that the elements $a$ and $ga{g}^{-1}$ are conjugate.

Proposition

Let $G$ be a group and $g\in G$. Then ${\gamma }_g:G\to G$ is an isomorphism.

Proof We first check that ${\gamma }_g$ is a homomorphism:${\gamma }_g(ab)=gab{g}^{-1}=ga(g^{-1}g)b{g}^{-1}=(ga{g}^{-1})(gb{g}^{-1})={\gamma }_g(a){\gamma }_g(b)$The map is injective because ${\gamma }_g(a)=e⟹ga{g}^{-1}=e⟹ga=g⟹a=e$. And is surjective since for all $a\in G$, we have ${\gamma }_g(g^{-1}ag)=a$.$\square$