Homomorphisms and normal subgroups

A homomorphism $f:G\to K$ satisfies $f(ab)=f(a)f(b)$ for all inputs. The two multiplication symbols refer to the operations in their respective groups; those operations need not be written in the same notation. For example, an exponential map converts addition of real numbers into multiplication of complex unit numbers. Merely assigning each input an output is not enough: the compatibility equation must hold for every pair, including pairs whose values are not convenient examples.

A homomorphism automatically sends the source identity to the target identity. The equation $f(e)=f(e)^2$ and cancellation prove this. Applying the map to $gg^{-1}=e$ then gives $f(g^{-1})=f(g)^{-1}$. These deductions make images of subgroups into subgroups: products and inverses of image elements have preimages obtained by performing the corresponding source operations. The whole image may be a proper subgroup of the declared target, so a homomorphism need not be surjective.

The kernel consists of all source elements mapping to the target identity. It is a subgroup, since the image of a product with an inverse is again the identity. More importantly, $f(a)=f(b)$ exactly when $a^{-1}b$ belongs to the kernel. Thus fibers of the map are left cosets of the kernel. The kernel is trivial precisely when no two distinct inputs share an output, which is the definition of injectivity.

Consider reduction of integers modulo five, from the additive integers to the additive residue group. The image contains every residue, while the kernel is the set of integer multiples of five. Inputs such as two and twelve agree after reduction because their difference belongs to that kernel. This illustrates the distinction between an input becoming zero and two arbitrary inputs becoming equal: the latter happens when their difference, rather than either input individually, becomes zero.

A subgroup $N$ is normal when conjugating by any ambient element preserves it: $gNg^{-1}=N$. A kernel always has this property because the image of $gng^{-1}$ becomes $f(g)e_Kf(g)^{-1}$ whenever $n$ lies in the kernel. In an abelian group, conjugation changes nothing, so every subgroup is normal. In a nonabelian group, normality is a genuine additional requirement, not a synonym for being a subgroup.

For example, the subgroup of $S_3$ containing the identity and the swap $(1\ 2)$ is not normal. Conjugating that swap by $(1\ 2\ 3)$ relabels its two entries and produces $(2\ 3)$, which is outside the subgroup. By contrast, the even permutations form the kernel of the sign map and are normal. Normality is exactly the ingredient that will allow multiplication of cosets without the result depending on which representatives were chosen.