A group remembers how its elements multiply, but mathematics rarely studies a group in isolation. We compare a symmetry group with a permutation group, reduce an infinite group modulo a [normal subgroup](/page/Normal%20Subgroup), or represent abstract elements as matrices, and the basic question is: what kind of function between groups respects the multiplication enough that group-theoretic information survives the trip?

A bijection of underlying sets is too weak. It may rename the elements while destroying products. A map that is not bijective may still be valuable, because it can forget exactly the information we want to discard. The useful notion is not sameness of sets, but compatibility with the group operation.

[example: Parity as a Multiplicative Shadow] Let $G=(\mathbb{Z},+)$ and let $H=\{1,-1\}$ under multiplication. Define \begin{align*} \varphi:\mathbb{Z}&\to H\\ n&\mapsto(-1)^n. \end{align*} This is a well-defined function into $H$, because $(-1)^n=1$ when $n$ is even and $(-1)^n=-1$ when $n$ is odd. For all $m,n\in\mathbb{Z}$, the integer exponent law gives \begin{align*} \varphi(m+n)&=(-1)^{m+n}\\ &=(-1)^m(-1)^n\\ &=\varphi(m)\varphi(n). \end{align*} Thus the operation in the domain is transported to the operation in the codomain: adding $m$ and $n$ in $\mathbb{Z}$ corresponds exactly to multiplying $(-1)^m$ and $(-1)^n$ in $H$. The map forgets everything about an integer except its parity, and that parity information is precisely what survives as the sign $1$ or $-1$. [/example]

This example is the model for the whole chapter. A homomorphism may compress, embed, relabel, or represent a group, but it always does so by respecting products. Kernels, images, quotient groups, and isomorphism theorems are the language for describing what is preserved and what is lost.

## Definition

### Preserving the Operation

The product in the domain and the product in the codomain may be written differently, so the definition must compare both group laws. The condition below is the minimum requirement that every multiplication computation in the domain can be transported into the codomain.

[definition: Group Homomorphism] Let $(G,\cdot_G)$ and $(H,\cdot_H)$ be groups. A group homomorphism from $G$ to $H$ is a function $\varphi:G\to H$ such that for all $g_1,g_2\in G$, \begin{align*} \varphi(g_1\cdot_G g_2)=\varphi(g_1)\cdot_H\varphi(g_2). \end{align*} [/definition]

After the operations are fixed, one usually writes $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$. The shorter notation is safe because the domain and codomain determine which product is being used.

The definition mentions only products, but a group also has a distinguished identity and inverse operation. At first this leaves a possible gap: a function might preserve multiplication while mishandling those extra pieces of structure. To use homomorphisms as structure-preserving maps, that ambiguity has to be removed.

[quotetheorem:4981]

The theorem removes a hidden worry in the definition. A homomorphism is allowed to mention only products because the identity and inverse operations are then forced to behave correctly. This is what lets group homomorphisms preserve equations, powers, orders, and relations without separately checking every piece of the group structure.

### Equations and Relations

A homomorphism does more than preserve individual products. It sends every equation built from multiplication and inverses in the domain to a corresponding equation in the codomain. This is why homomorphisms are useful for studying groups given by relations.

[example: Relations Survive Under Homomorphisms] Let $\varphi:G\to H$ be a group homomorphism, and suppose $g\in G$ satisfies $g^6=1_G$. We compute the sixth power of $\varphi(g)$ by repeatedly using the homomorphism property: \begin{align*} \varphi(g)^2&=\varphi(g)\varphi(g)=\varphi(g^2),\\ \varphi(g)^3&=\varphi(g)^2\varphi(g)=\varphi(g^2)\varphi(g)=\varphi(g^3),\\ \varphi(g)^4&=\varphi(g)^3\varphi(g)=\varphi(g^3)\varphi(g)=\varphi(g^4),\\ \varphi(g)^5&=\varphi(g)^4\varphi(g)=\varphi(g^4)\varphi(g)=\varphi(g^5),\\ \varphi(g)^6&=\varphi(g)^5\varphi(g)=\varphi(g^5)\varphi(g)=\varphi(g^6). \end{align*} Since $g^6=1_G$, this gives \begin{align*} \varphi(g)^6=\varphi(g^6)=\varphi(1_G)=1_H, \end{align*} where the last equality uses *[Homomorphisms Preserve Identity and Inverses](/theorems/4981)*. Thus $\varphi(g)$ satisfies $x^6=1_H$, so the order of $\varphi(g)$ divides $6$. A homomorphism may reduce the order of an element, but it cannot send a solution of $x^6=1_G$ to an element that violates the corresponding equation in $H$. [/example]

This relation-preserving property is the practical reason homomorphisms appear everywhere in algebra. They allow us to replace a group by a more concrete or smaller one while keeping control of all multiplicative identities.

## Kernels and Images

The first diagnostic for a homomorphism is to ask which elements become indistinguishable from the identity. Those elements are precisely the information lost by the map, and collecting them turns a question about a function into a question about a subgroup.

[definition: Kernel of a Group Homomorphism] Let $\varphi:G\to H$ be a group homomorphism. The kernel of $\varphi$ is \begin{align*} \ker\varphi=\{g\in G:\varphi(g)=1_H\}. \end{align*} [/definition]

The kernel of the parity homomorphism is $2\mathbb{Z}$. More generally, the kernel records the identity fibre of the homomorphism, and all other fibres are its cosets. To use kernels in quotient constructions, we need them to be normal subgroups rather than arbitrary subgroups.