In this lecture, we study maps from a group into another group that carry over the same group structure i.e preserve group operations. Such maps are called homomorphisms. It turns out that it suffices to require a homomorphism to preserve a binary operation only. Then the unary operation (inverses) and the nullary operation (the identity element) are automatically preserved.

*Definition*. Let $(G,\ast)$ and $(G’,\sharp)$ be two groups. A map $\varphi: G\longrightarrow G’$ is called a *homomorphism* if $\varphi(a\ast b)=\varphi(a)\sharp\varphi(b)$.

*Example*. Define $\varphi: (\mathbb{R}^+,\cdot)\longrightarrow(\mathbb{R},+)$ by

$$\varphi(x)=\log_{10}x,\ \forall x\in\mathbb{R}^+.$$

Then $\varphi$ is a homomorphism. In addition, $\varphi$ is one-to-one and onto. A homomorphism which is one-to-one and onto is called an *isomorphism*. If there is an isomorphism from a group onto another group, they are the same group meaning we do not distinguish them.

*Example*. The map $\varphi: \mathbb{Z}\longrightarrow\mathbb{Z}_n$ defined by

$$\varphi(x)=[x],\ \forall\ x\in\mathbb{Z}$$

is a homomorphism called the *canonical* or the *natural homomorphism*. The canonical homomorphism is onto.

*Example*. Let $G$ be a group and $A(G)$ the set of all bijective (i.e. one-to-one and onto) maps of $G$ onto itself. Recall that $A(G)$ forms a group with composition. Let $a\in G$. Define $T_a: G\longrightarrow G$ by

$$T_ax=ax\ \forall\ x\in G.$$ Hence $\forall a\in G$, $T_a\in A(G)$. Furthermore, $\forall a,b\in G$, $T_aT_b=T_{ab}$: $\forall x\in G$,

$$T_aT_b(x)=T_a(T_bx)=T_a(bx)=a(bx)=(ab)x=T_{ab}x.$$

Define $\varphi: G\longrightarrow A(G)$ by

$$\varphi(a)=T_a,\ \forall\ a\in G.$$ Then $\varphi$ is a homomorphism. In addition, it is one-to-one. If $|G|=n$, then $|A(G)|=n!$, so $\varphi$ cannot be onto.

*Definition*. A homomorphism which is one-to-one (injective) is called a *monomorphism*. A homomorphism which is onto (surjective) is called an *epimorphism*. A homomorphism which is one-to-one and onto (bijective) is called an *isomorphism* as mentioned above. $G$ is said to beĀ *isomorphic* to $G’$ if there exists an isomorphism from $G$ onto $G’$. If $G$ is isomorphic to $G’$, we write $G\cong G’$. An isomorphism from a group $G$ onto itself is called an *automorphism*.

*Theorem*. [Cayley’s Theorem] Every group $G$ is isomorphic to some subgroup of $A(S)$, for an appropriate set $S$.

We in fact proved this theorem in the previous example by taking $S$ to be the group $G$ itself. But there may be other choices for $S$. If $G$ is finite, $S$ may be taken to be a finite set in which case $A(S)$ is $S_n$, the group of permutations of $\{1,2,\cdots,n\}$. In this case, Cayley’s Theorem is stated as: A finite group can be represented as a group of permutations.

*Example*. Let $G$ be a group and $a\in G$ be fixed. The map $\varphi: G\longrightarrow G$ defined by

$$\varphi(x)=a^{-1}xa,\ \forall x\in G$$

is an automorphism called the *inner automorphism* of $G$ induced by $a$.

*Example*. Define $\varphi: (\mathbb{R},+)\longrightarrow (\mathbb{C}\setminus\{0\},\cdot)$ by

$$\varphi(x)=e^{ix},\ \forall x\in\mathbb{R}.$$

While $\varphi$ is a homomorphism, it is neither one-to-one nor onto.

*Lemma*. Let $\varphi: G\longrightarrow G’$ be a homomorphism. Then

(a) $\varphi(e)=e’$.

(b) $\varphi(a^{-1})=\varphi(a)^{-1}$.

*Proof*. (a) Let $a\in G$. Then $ae=a$. Since $\varphi$ is a homomorphism, $$\varphi(ae)=\varphi(a)\varphi(e)=\varphi(a)=\varphi(a)e’.$$ So by the cancellation law, we obatin $\varphi(e)=e’$.

(b) Let $a\in G$. Then

\begin{align*}

\varphi(a)\varphi(a^{-1})&=\varphi(aa^{-1})\ (\mbox{because $\varphi$ is a homomorphism})\\

&=\varphi(e)\\

&=e’\ (\mbox{by part (a)})

\end{align*}

Hence, $\varphi(a)^{-1}=\varphi(a^{-1})$.

*Lemma*. Let $\varphi: G\longrightarrow G’$ be a homomorphism. Then $\varphi(G)\leq G’$.

In here, the kernel of a function was introduced. Let $\varphi: G\longrightarrow G’$ be a homomorphism and $K=\varphi^{-1}(e’)=\{a\in G: \varphi(a)=e’\}$. Then $\ker\varphi$ and $K=\varphi^{-1}(e’)$, the preimage of $e’$ under $\varphi$ are closely related as:

\begin{align*}

(a,b)\in\ker\varphi&\Longleftrightarrow\varphi(a)=\varphi(b)\\

&\Longleftrightarrow\varphi(ab^{-1})=e’\\

&\Longleftrightarrow ab^{-1}\in K.

\end{align*}

So, in group theory we define $K=\varphi^{-1}(e’)$ to be $\ker\varphi$, the kernel of $\varphi$.

*Theorem*. $\varphi: G\longrightarrow G’$ is one-to-one if and only if $\ker\varphi=\{e\}$.

*Proof*. ($\Rightarrow$) Suppose that $\varphi$ is one-to-one and let $a\in\ker\varphi$. Then $\varphi(a)=\varphi(e)=e’$. Since $\varphi$ is one-to-one, $a=e$.

($\Leftarrow$) Suppose that $\ker\varphi=\{e\}$ and let $\varphi(a)=\varphi(b)$. Then

\begin{align*}

\varphi(ab^{-1})=e’&\Longrightarrow ab^{-1}\in\ker\varphi\\

&\Longrightarrow ab^{-1}=e\\

&\Longrightarrow a=b.

\end{align*}

So, $\varphi$ is one-to-one.

*Theorem*. Let $\varphi: G\longrightarrow G’$ be a homomorphism. Then $\ker\varphi\leq G$.

*Theorem*. Let $K=\ker\varphi$. Then $\forall a\in G$, $aK=Ka$.

*Proof*. It suffices to show (why?) that $\forall a\in G$, $a^{-1}Ka\subset K$. Let $a\in G$. Then $\forall k\in K$,

$$\varphi(a^{-1}ka)=\varphi(a)^{-1}\varphi(k)\varphi(a)=e’\Longrightarrow a^{-1}ka\in K.$$