Section3.2Definitions of Homomorphisms and Isomorphisms¶ permalink

Definition3.2.1

Let $\langle S,*\rangle$ and $\langle S',*'\rangle$ be binary structures. A function $\phi$ from $S$ to $S'$ is a homomorphism if

\begin{equation*} \phi(a* b)=\phi(a)*'\phi(b) \end{equation*}

for all $a,b\in S\text{.}$ An isomorphism is a homomorphism that is also a bijection.

Intuitively, you can think of a homomorphism $\phi$ as a “structure-preserving” map: if you multiply and then apply $\phi\text{,}$ you get the same result as when you first apply $\phi$ and then multiply. Isomorphisms, then, are both structure-preserving and cardinality-preserving.

Note3.2.2

We may omit the $*$ and $*'\text{,}$ as per our group conventions, but we include them here to emphasize that the operations in the structures may be distinct from one another. When we omit them and write $\phi(st)=\phi(s)\phi(t)\text{,}$ then it is the writers' and readers' responsibility to keep in mind that $s$ and $t$ are being operated together using the operation in $S\text{,}$ while $\phi(s)$ and $\phi(t)$ are being operated together using the operation in $S'\text{.}$

Remark3.2.3

There may be more than one homomorphism [isomorphism] from one binary structure to another (see Example 3.2.4).

Example3.2.4

For each of the following, decide whether or not the given function $\phi$ from one binary structure to another is a homomorphism, and, if so, if it is an isomorphism. Prove or disprove your answers! For Parts 6 and 7, $C^0$ is the set of all continuous functions from $\R$ to $\R\text{;}$ $C^1$ is the set of all differentiable functions from $\R$ to $\R$ whose derivatives are continuous; and each $+$ indicates pointwise addition on $C^0$ and $C^1\text{.}$

1. $\phi:\langle \Z,+\rangle \to \langle \Z,+\rangle$ defined by $\phi(x)=x\text{;}$

2. $\phi:\langle \Z,+\rangle \to \langle \Z,+\rangle$ defined by $\phi(x)=-x\text{;}$

3. $\phi:\langle \Z,+\rangle \to \langle \Z,+\rangle$ defined by $\phi(x)=2x\text{;}$

4. $\phi:\langle \R,+\rangle \to \langle \R^+,\cdot\,\rangle$ defined by $\phi(x)=e^x\text{;}$

5. $\phi:\langle \R,+\rangle \to \langle \R^*,\cdot\,\rangle$ defined by $\phi(x)=e^x\text{;}$

6. $\phi:\langle C^1,+\rangle \to \langle C^0,+\rangle$ defined by $\phi(f)=f'$ (the derivative of $f$);

7. $\phi:\langle C^0,+\rangle \to \langle \R,+\rangle$ defined by $\phi(f)=\displaystyle{\int_0^1 f(x)\, dx}\text{.}$

Example3.2.5

Let $\Gdot$ be a group and let $a\in G\text{.}$ Then the function $c_a$ from $G$ to $G$ defined by $c_a(x)=axa^{-1}$ (for all $x\in G$) is a homomorphism. Indeed, let $x,y\in G\text{.}$ Then

\begin{align*} c_a(xy)\amp =a(xy)a^{-1}\\ \amp =(ax)e(ya^{-1})\\ \amp =(ax)(a^{-1}a)(ya^{-1})\\ \amp =(axa^{-1})(aya^{-1})\\ \amp =c_a(x)c_a(y). \end{align*}

The homomorphism $c_a$ is called conjugation by $a$.

Definition3.2.6

Homomorphisms from a group $G$ to itself are called endomorphisms, and isomorphisms from a group to itself are called automorphisms.

It can be shown that conjugation by any element $a$ of a group $G$ is a bijection from $G$ to itself (can you prove this?), so such conjugation is an automorphism of $G\text{.}$ (Beware: Some texts use “conjugation by $a$” to refer to the function $x\mapsto a^{-1}xa\text{.}$) Both versions of conjugation by $a$ in group $G$ are automorphisms of $G\text{.}$)

We end with a theorem stating basic facts about homomorphisms from one group to another. (Note. This doesn't apply to arbitrary binary structures, which may or may not even have identity elements.)