When we want to compare two groups, or compress one group into a simpler one, the right tool is a map that respects the group structure. The definition asks for surprisingly little:

That is a single equation. But a group carries three pieces of structure – a binary operation, an identity element, and inverses. Should we not also require $\phi (e_G)=e_H$ and $\phi (a^{-1})=\phi (a{)}^{-1}$? It seems as though we are being careless by imposing only one condition out of a possible three.

In fact, the single condition is far more powerful than it appears. The other two are automatic consequences.

1 One condition entails the rest

The preservation of products, $\phi (ab)=\phi (a)\phi (b)$, forces the identity and inverses to be preserved as well.

Since $e_G=e_G\cdot e_G$, applying $\phi$ yields $\phi (e_G)=\phi (e_G)\phi (e_G)$. Multiplying both sides on the left by $\phi (e_G{)}^{-1}$ gives $e_H=\phi (e_G)$.

For inverses: from $a\cdot a^{-1}=e_G$ we get $\phi (a)\phi (a^{-1})=\phi (e_G)=e_H$, so $\phi (a^{-1})$ is the inverse of $\phi (a)$ in $H$. Since inverses in a group are unique, $\phi (a^{-1})=\phi (a{)}^{-1}$.

2 A catalog of homomorphisms

Definitions come alive through examples. Here is a tour of the most important homomorphisms in algebra.

The natural projection$\pi :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$. Send each integer to its residue class modulo $n$: $\pi (a)=a\mathrm{mod}n$. The identity $(a+b)\mathrm{mod}n=(a\mathrm{mod}n)+(b\mathrm{mod}n)$ is precisely the homomorphism condition. This is the most elementary example, and it is the quotient-group construction from the previous article in disguise.

The sign homomorphism$\mathrm{sgn}:S_n\to \{+1,-1\}$. Assign $+1$ to every even permutation and $-1$ to every odd one. The multiplicativity $\mathrm{sgn}(\sigma \tau )=\mathrm{sgn}(\sigma )\cdot \mathrm{sgn}(\tau )$ makes this a homomorphism. It compresses the $n!$ elements of $S_n$ down to just two values.

The determinant$\det :G{L}_n(\mathbb{R})\to {\mathbb{R}}^{\ast }$. The familiar identity $\det (AB)=\det (A)\det (B)$ from linear algebra is exactly the homomorphism condition, read through the lens of group theory. A matrix carrying an enormous amount of data is reduced to a single nonzero real number.

The exponential$\exp :(\mathbb{R},+)\to ({\mathbb{R}}_{>0},\times )$. The law of exponents $\exp (a+b)=\exp (a)\cdot \exp (b)$ makes the exponential function a homomorphism from the additive reals to the positive multiplicative reals. Its inverse $\log$ is also a homomorphism: $\log (xy)=\log (x)+\log (y)$.

Complex conjugation$\overline{}:({\mathbb{C}}^{\ast },\times )\to ({\mathbb{C}}^{\ast },\times )$. The identity $\overline{zw}=\bar{z}\bar{w}$ makes conjugation a homomorphism, and since $\overline{\bar{z}}=z$ (so the map is its own inverse), it is in fact an isomorphism.

The trivial homomorphism. For any groups $G$ and $H$, the map $\phi (g)=e_H$ is a homomorphism. It sends everything to the identity – the worst possible compression, destroying all information – but it does satisfy the condition.

The inclusion map. If $H\le G$ is a subgroup, the inclusion $\iota :H\hookrightarrow G$ defined by $\iota (h)=h$ is an injective homomorphism: it preserves every detail.

3 The kernel: what gets crushed

The kernel of a homomorphism $\phi :G\to H$ is the set of elements that $\phi$ sends to the identity of $H$:

Let us identify the kernel in each of our examples:

• $\pi :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$:$\ker =n\mathbb{Z}$ (the multiples of $n$)

• $\mathrm{sgn}:S_n\to \{+1,-1\}$:$\ker =A_n$ (the even permutations – the alternating group)

• $\det :G{L}_n(\mathbb{R})\to {\mathbb{R}}^{\ast }$:$\ker =S{L}_n(\mathbb{R})$ (matrices with determinant $1$)

• $\exp :\mathbb{R}\to {\mathbb{R}}_{>0}$:$\ker =\{0\}$ (the exponential is injective)

• trivial homomorphism:$\ker =G$ (everything is crushed)

• inclusion $H\hookrightarrow G$:$\ker =\{e\}$ (nothing is lost)

The kernel measures how much information the homomorphism destroys. A large kernel means heavy compression; a trivial kernel $\ker \phi =\{e_G\}$ means $\phi$ is injective.

4 The kernel is always a normal subgroup

Here is a fact of fundamental importance: the kernel of any group homomorphism is a normal subgroup of the domain.

Take $n\in \ker \phi$ and any $g\in G$. Then

Revisiting our examples: $n\mathbb{Z}⊴\mathbb{Z}$, $A_n⊴S_n$, $S{L}_n(\mathbb{R})⊴G{L}_n(\mathbb{R})$– all of these are well-known normal subgroups. But their normality does not need to be verified by a separate argument; it follows in a single stroke from the fact that each is the kernel of a homomorphism.

And the converse holds as well: every normal subgroup$N⊴G$is the kernel of some homomorphism. The natural projection $\pi :G\to G/N$ defined by $g\mapsto gN$ is a surjective homomorphism whose kernel is exactly $N$.

This means that "normal subgroup" and "kernel of a homomorphism" are two descriptions of the same concept. The previous article showed that normality is the condition needed for the quotient construction to work; now we see that normality is also the condition that arises whenever a homomorphism compresses a group. The two perspectives are complementary faces of a single idea.

5 The image: how far the map reaches

The image of $\phi$ is the subset of $H$ that $\phi$ actually hits:

In our examples:

• $\pi :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$:$\mathrm{Im}=\mathbb{Z}/n\mathbb{Z}$ (surjective)

• $\mathrm{sgn}$:$\mathrm{Im}=\{+1,-1\}$ (surjective for $n\ge 2$)

• $\det$:$\mathrm{Im}={\mathbb{R}}^{\ast }$ (surjective)

• $\exp$:$\mathrm{Im}={\mathbb{R}}_{>0}$ (does not reach the negative reals – $e^x$ is always positive)

The image is always a subgroup of $H$. When it equals $H$, the homomorphism is surjective.

6 Injectivity, surjectivity, and isomorphisms

The kernel and image give clean, algebraic criteria for the fundamental set-theoretic properties of a map:

• $\phi$ is injective$⟺$$\ker \phi =\{e_G\}$.

• $\phi$ is surjective$⟺$$\mathrm{Im}\phi =H$.

• When both conditions hold, $\phi$ is called an isomorphism, and we write $G\cong H$.

The injectivity criterion is remarkably efficient: instead of checking that $\phi (a)\ne \phi (b)$ for every pair of distinct elements, we need only verify that the kernel is trivial. For instance, $\exp :(\mathbb{R},+)\to ({\mathbb{R}}_{>0},\times )$ has $\ker =\{0\}$ (injective) and $\mathrm{Im}={\mathbb{R}}_{>0}$ (surjective), so it is an isomorphism: the additive group $(\mathbb{R},+)$ and the multiplicative group $({\mathbb{R}}_{>0},\times )$ are the same group in different clothing. Surprising at first glance, but entirely natural once you recall that $\exp$ and $\log$ convert between addition and multiplication.

7 Takeaway

The single condition $\phi (ab)=\phi (a)\phi (b)$ turns out to encode everything: preservation of the identity, preservation of inverses, and the deep structural link between kernels and normal subgroups. The kernel captures what the homomorphism cannot see; the image captures what it can reach. Together they determine the homomorphism up to the isomorphism that is the subject of the next article: the First Isomorphism Theorem.