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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section3.2Definitions of Homomorphisms and Isomorphisms


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.


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 \(S'\text{.}\)


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


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 \(\phi(f)=\displaystyle{\int_0^1 f(x)\, dx}\text{.}\)


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 \(a\).


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.)