Once a function has passed the homomorphism test, we can ask whether it is onto, one-to-one, or both. The focus keyword epimorphism monomorphism isomorphism automorphism collects the four standard names used for these special cases. A homomorphism may preserve the operation while still losing information or missing part of the codomain. A monomorphism preserves distinctness, an epimorphism reaches every codomain element, and an isomorphism does both. An automorphism is an isomorphism from a group to itself. These words are not decorative vocabulary; they describe exactly how much structure the map preserves and how completely it represents one group inside another. :::definition[Epimorphism] Let $f:G\to G_1$ be a homomorphism. Then $f$ is called an epimorphism if $f$ is onto. Equivalently, $$\begin{equation*} f(G)=G_1. \end{equation*}$$ ::: :::definition[Monomorphism] Let $f:G\to G_1$ be a homomorphism. Then $f$ is called a monomorphism if $f$ is one-to-one. Equivalently, for all $a,b\in G$, $$\begin{equation*} f(a)=f(b)\implies a=b. \end{equation*}$$ ::: :::definition[Isomorphism] Let $f:G\to G_1$ be a homomorphism. Then $f$ is called an isomorphism if $f$ is both one-to-one and onto. If there exists an isomorphism from $G$ onto $G_1$, then $G$ and $G_1$ are called isomorphic groups, written $$\begin{equation*} G\cong G_1. \end{equation*}$$ ::: :::definition[Automorphism] Let $G$ be a group. An automorphism of $G$ is an isomorphism from $G$ onto $G$. Thus an automorphism preserves the group operation and rearranges the elements of $G$ without leaving the group. ::: Two isomorphic groups may use different symbols, different elements, and different operations in appearance, but they have the same algebraic structure. The isomorphism pairs each element of one group with exactly one element of the other group and respects multiplication. Students often say that isomorphic groups are equal. It is safer to say that they are structurally the same. Equality compares objects literally; isomorphism compares group structure. :::example[Epimorphism That Is Not a Monomorphism] Let $n>1$ and define $f:(\mathbb{Z},+)\to(\mathbb{Z}_n,+_n)$ by $f(a)=[a]$. [1] The map is a homomorphism because $$\begin{equation*} f(a+b)=[a+b]=[a]+_n[b]=f(a)+_n f(b). \end{equation*}$$ [2] The map is onto because every class $[r]\in\mathbb{Z}_n$ is the image of $r\in\mathbb{Z}$. [3] The map is not one-to-one because $f(0)=[0]=[n]=f(n)$ but $0\neq n$. Hence $f$ is an epimorphism but not a monomorphism. $\square$ ::: :::example[Monomorphism That Is Not an Epimorphism] Let $f:(\mathbb{Z},+)\to(\mathbb{Z},+)$ be defined by $f(a)=2a$. [1] The map is a homomorphism because $f(a+b)=2a+2b=f(a)+f(b)$. [2] If $f(a)=f(b)$, then $2a=2b$, so $a=b$. Hence $f$ is one-to-one. [3] The map is not onto because no integer $a$ satisfies $2a=1$. Hence $f$ is a monomorphism but not an epimorphism. $\square$ ::: :::example[Exponential Isomorphism] Define $f:(\mathbb{R},+)\to(\mathbb{R}^{+},\cdot)$ by $f(a)=2^a$. [1] For $a,b\in\mathbb{R}$, $$\begin{equation*} f(a+b)=2^{a+b}=2^a2^b=f(a)f(b), \end{equation*}$$ so $f$ is a homomorphism. [2] If $f(a)=f(b)$, then $2^a=2^b$, so $a=b$. Thus $f$ is one-to-one. [3] If $y\in\mathbb{R}^{+}$, then $y=2^{\log_2 y}$, so $y=f(\log_2 y)$. Thus $f$ is onto. Hence $f$ is an isomorphism and $(\mathbb{R},+)\cong(\mathbb{R}^{+},\cdot)$. $\square$ ::: :::solved-problem Let $f:\mathbb{R}^{\ast}\to\mathbb{R}^{\ast}$ be defined by $f(a)=|a|$, where $\mathbb{R}^{\ast}=\mathbb{R}\setminus\{0\}$ under multiplication. Determine whether $f$ is a homomorphism, monomorphism, or epimorphism. ::: :::solution Let $a,b\in\mathbb{R}^{\ast}$. Then $$\begin{align*} f(ab)&=|ab|\\ &=|a||b|\\ &=f(a)f(b). \end{align*}$$ Therefore $f$ is a homomorphism. Since $$\begin{equation*} f(1)=1=f(-1) \end{equation*}$$ but $1\neq -1$, the map is not one-to-one. Hence it is not a monomorphism. The image of $f$ is $\mathbb{R}^{+}$, so no negative real number lies in the image. Therefore $f$ is not onto $\mathbb{R}^{\ast}$. Hence it is not an epimorphism. Thus $f$ is a homomorphism, but it is neither a monomorphism nor an epimorphism. $\square$ ::: :::tip A homomorphism must be checked before applying the special names. A one-to-one function that does not preserve the operation is not a monomorphism of groups. An onto function that does not preserve the operation is not an epimorphism of groups. The words epimorphism monomorphism isomorphism automorphism always start with the homomorphism condition. ::: :::exercise 1. Decide whether $f:(\mathbb{Z},+)\to(\mathbb{Z},+)$ defined by $f(n)=-n$ is an automorphism. 2. Give an example of a homomorphism that is neither one-to-one nor onto. 3. If $f:G\to H$ is an isomorphism, what are $\ker f$ and $f(G)$? ::: :::answer 1. Yes. It is a homomorphism, one-to-one, and onto, so it is an automorphism of $\mathbb{Z}$. 2. The map $f:(\mathbb{Z},+)\to(\mathbb{Z},+)$ defined by $f(n)=0$ is neither one-to-one nor onto. 3. Since $f$ is one-to-one, $\ker f=\{e_G\}$. Since $f$ is onto, $f(G)=H$. ::: :::faq[Frequently Asked Questions] Q: Is every isomorphism an epimorphism? A: Yes. An isomorphism is onto, so it is an epimorphism. Q: Is every automorphism an isomorphism? A: Yes. An automorphism is an isomorphism from a group onto itself. Q: Can a homomorphism be both a monomorphism and an epimorphism? A: Yes. In that case it is an isomorphism. ::: :::call-to-action[Continue to Natural Homomorphisms] subtitle: See how every normal subgroup appears as the kernel of a canonical homomorphism onto a quotient group. button: Study Natural Homomorphisms | /abstract-algebra/homomorphisms-and-isomorphisms-of-groups/homomorphisms-of-groups/natural-homomorphism button-ghost: Review Kernels | /abstract-algebra/homomorphisms-and-isomorphisms-of-groups/homomorphisms-of-groups/kernel-of-a-homomorphism :::