A group homomorphism \varphi : G \to H is a structure-preserving map, but it is usually neither injective nor surjective: it can squash several elements of G onto the same target, and it need not hit all of H. Two pieces of information measure exactly how far it strays from being a perfect matching — its kernel (what it collapses) and its image (what it reaches).

The First Isomorphism Theorem is the single most useful sentence in all of group theory, because it says these two measurements are secretly the same thing. Collapse the kernel to a point — that is, pass to the quotient group G/\ker\varphi — and what remains is a flawless copy of the image:

This one line is a factory for producing isomorphisms. Any time you can build a homomorphism out of a group, its kernel and image hand you a quotient you already understand — no clever guessing required. This page is about that theorem, taught slowly and from several angles; the Second, Third and Correspondence theorems appear near the end as its natural companions.

The idea: injectivity is failure to be one-to-one, and the kernel records it

Recall the two fingerprints of a homomorphism \varphi : G \to H:

• the kernel \ker\varphi = \{\, g \in G : \varphi(g) = e_H \,\} — a normal subgroup of G;
• the image \operatorname{im}\varphi = \{\, \varphi(g) : g \in G \,\} — a subgroup of H.

Here is the key observation. Two elements a, b \in G have the same image, \varphi(a) = \varphi(b), exactly when \varphi(a b^{-1}) = e_H, i.e. when a and b differ by an element of the kernel — precisely when they lie in the same coset of \ker\varphi. So \varphi fails to be injective in exactly the amount described by its kernel: everything in one coset of \ker\varphi is fused to a single point of the image.

Now do the natural thing: glue each coset to a point. The cosets of \ker\varphi are the elements of the quotient group G/\ker\varphi, and on that quotient \varphi can no longer collapse anything — distinct cosets map to distinct images. The induced map is therefore injective, and it is obviously onto the image, so it is a bijective homomorphism: an isomorphism. That is the whole theorem.

The factorisation picture

The cleanest way to see the theorem is as a factorisation. Every homomorphism \varphi : G \to H breaks into three honest steps:

1. \pi, the quotient map (or "projection"), which is surjective and does all the collapsing, sending g \mapsto g\ker\varphi;
2. the induced map \bar\varphi, which is an isomorphism onto the image and does no collapsing at all;
3. the inclusion \operatorname{im}\varphi \hookrightarrow H, which is injective and merely remembers that the image sits inside H.

Surjection, then isomorphism, then injection: every homomorphism is that sandwich. The diagram below builds the picture step by step — press play, and watch the middle arrow become the equality the theorem promises.

The universal property, said plainly

"Unique" is the payoff: the isomorphism is not something you have to construct by hand and hope works — it is forced. Once you know the kernel and the image, the copy is determined, and the diagram commutes on its own. This is why the theorem is used so casually: to identify a quotient, you just find a homomorphism whose kernel is the subgroup you divided by, and read off the image.

Worked example 1: \det : GL_n(\mathbb{R}) \to \mathbb{R}^\times

The determinant is a homomorphism from invertible matrices under multiplication to nonzero reals under multiplication, because \det(AB) = \det(A)\det(B). It is surjective — every nonzero real is the determinant of some diagonal matrix — so its image is all of \mathbb{R}^\times. Its kernel is the set of matrices with determinant 1, which is the special linear group SL_n(\mathbb{R}). The First Isomorphism Theorem instantly delivers

Read that as a sentence: "invertible matrices, once you forget everything except their determinant, are the nonzero reals." We proved a nontrivial isomorphism without exhibiting a single explicit bijection between cosets and reals — the theorem did the labour.

Worked example 2: the sign map \operatorname{sgn} : S_n \to \{\pm 1\}

The sign (or parity) homomorphism sends each permutation to +1 if it is even and -1 if it is odd, and it respects composition: the sign of a product is the product of the signs. For n \ge 2 it is surjective onto the two-element group \{\pm 1\}, and its kernel — the even permutations — is precisely the alternating group A_n. Therefore

In particular the index [S_n : A_n] = 2: exactly half of all permutations are even. The theorem turns a counting fact about parity into a clean structural statement.

Worked example 3: reduction \mathbb{Z} \to \mathbb{Z}_n, and a circle

Let \varphi : \mathbb{Z} \to \mathbb{Z}_n send each integer to its remainder modulo n. This is a surjective homomorphism, and an integer maps to 0 exactly when it is a multiple of n, so \ker\varphi = n\mathbb{Z}. Out drops the very definition of modular arithmetic:

A continuous cousin: the map \varphi : \mathbb{R} \to S^1, \theta \mapsto e^{\,i\theta}, is a homomorphism from the reals under addition to the unit circle under multiplication. It is onto, and e^{\,i\theta} = 1 exactly when \theta is a multiple of 2\pi, so \ker\varphi = 2\pi\mathbb{Z}. Hence

The real line, wrapped so that every full turn returns to the start, is the circle. Same theorem, same three moves — collapse the kernel, land on the image — whether the groups are finite, infinite, or continuous.

The other three theorems, briefly

Once the First Iso Theorem is in hand, three companions follow. They are stated here for completeness; each is really a corollary of the first, proved by building the right homomorphism and reading off its kernel.

It is called the diamond theorem because the four subgroups HN, H, N, H\cap N sit at the corners of a little diamond-shaped lattice, and the two "parallel" edges of the diamond represent isomorphic quotients.

This is the "cancellation" law: quotienting by N and then again by M/N is the same as quotienting by M in one go — the N's formally cancel, just like fractions.

In words: the lattice of subgroups of a quotient is a faithful copy of the top slice of the original's lattice — everything above N, carried down unchanged.