Definition. (Page 100) If $G$ is a group with operation $*$ and $H$ is a group with operation $\hash$ then a mapping $\theta:G\rightarrow H$ is a homomorphism if \[ \theta(a*b)=\theta(a)\hash\theta(b) \] for all $a,b\in G.$

Note. (Page 100) Every isomorphism is a homomorphism, but a homomorphism need not be one-to-one, and it need not be onto.

Example 21.1 For any positive integer $n$, define $\theta:\Z\rightarrow\Z_n$ by $\theta(a)=[a]$ for each $a\in\Z.$ Then $\theta(a+b)=[a+b]=[a]\oplus[b]=\theta(a)\oplus\theta(b)$ for all $a,b\in\Z,$ so that $\theta$ is a homomorphism. It is onto but not one-to-one.

Example 21.2 Define $\theta:\Z\rightarrow\Z$ by $\theta(a)=2a$ for each $a\in\Z.$ Then $\theta(a+b)=2(a+b)=2a+2b=\theta(a)+\theta(b)$ for all $a,b\in\Z.$ Thus $\theta$ is a homomorphism. It is one-to-one but not onto.

Example 21.3 For each $r\in\R,$ define $\rho_r:\R\rightarrow\R$ by $\rho_r(a)=ar$ for each $a\in\R.$ Then $\rho_r$ is a homomorphism from the additive group of $\R$ itself: $\rho_r(a+b)$ $=(a+b)r$ $=ar+br$ $=\rho_r(a)+\rho_r(b)$ for all $a,b\in\R.$ Notice that $\rho_r$ is a homomorphism precisely because of the distributive law $(a+b)r=ar+br.$

Example 21.4 Let $A$ and $B$ be groups, and $A\times B$ their direct product (defined in Section 15). Then $\pi_1:A\times B\rightarrow A$ defined by $\pi_1((a,b))=a$ is a homomorphism from $A\times B$ onto $A:$ $$\pi_1((a_1,b_1))=\pi_1((a_1a_2,b_1b_2))=a_1a_2=\pi_1((a_1,b_1))\pi_1((a_2,b_2))$$ (Problem 21.1 asks you to justify each step.) Also, $\pi_2:A\times B\rightarrow B$ defined by $\pi_2((a,b))=b$ is a homomorphism (Problem 21.1).

Note. (Page 101) Review Theorem 18.2 for some standard properties of homomorphisms.

Definition. (Page 101) The subgroup $\theta(G)$ of is called a homomorphic image of G. We shall see that nearly everything about a homomorphic image is determined by the domain of the homomorphism and the following subset of the domain.

Definition. (Page 101) If $\theta:G\rightarrow H$ is a homomorphism, then the kernel of $\theta$ is the set of all elements $a\in G$ such that $\theta(a)=e_H.$ This set will be denoted by $\Ker\theta.$ The kernel of a homomorphism is always a subgroup of the domain. Before proving that, however, let us look at some examples.

Example 21.5

Example 21.6

Example 21.7

Theorem 21.1 If $\theta:G\rightarrow H$ is a homomorphism, then $\Ker\theta$ is a subgroup of $G.$ Moreover, $\theta$ is one-to-one iff $\Ker\theta=\{e_G\}.$

Note. (Page 102) Kernels have one more property in common: they are all normal, in the sense of the next definition.

Definition. (Page 102) A subgroup $N$ of a group $G$ is a normal subgroup of $G$ if $gng^{-1}\in N$ for all $n\in N$ and all $g\in G.$ If $N$ is a normal subgroup, we write $N\lhd G.$

Example 21.8

Example 21.9

Note. (Page 102) A large collection of examples of normal subgroups--in fact, all examples, as we shall see in the next section--is given by the following theorem.

Theorem 21.2 If $G$ and $H$ are groups and $\theta:G\rightarrow H$ is a homomorphism, then $\Ker\theta\lhd G.$