First isomorphism theorem

The idea is that a homomorphism \(\varphi:G\to H\) identifies precisely those elements that differ by an element of the kernel: \(\varphi(a)=\varphi(b)\) is equivalent to \(a^{-1}b\in\ker\varphi\), that is, to \(a\) and \(b\) lying in the same coset. Hence the assignment \(a\ker\varphi\mapsto\varphi(a)\) is well defined and injective; it is a homomorphism because \(\varphi\) is, and it is onto the image by construction.

In practice the theorem is used in two ways. To identify a quotient: if you find a surjective homomorphism \(G\to H\) with kernel \(N\), then \(G/N\cong H\). Thus \(\det:GL_2(\mathbb{R})\to\mathbb{R}^*\) gives \(GL_2(\mathbb{R})/SL_2(\mathbb{R})\cong\mathbb{R}^*\), and reduction \(\mathbb{Z}\to\mathbb{Z}_n\) gives \(\mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}_n\). Or to count: \(|\operatorname{im}\varphi|=|G|/|\ker\varphi|\), so the image of \(x\mapsto 3x\) in \(\mathbb{Z}_{12}\) has \(12/3=4\) elements.

A limiting case helps fix ideas. If \(\varphi\) is injective, the kernel is trivial and \(G\cong\operatorname{im}\varphi\): \(G\) lives as a subgroup inside \(H\). If \(\varphi\) is trivial, the kernel is all of \(G\) and the quotient \(G/G\) is trivial, like the image. In general the image is a subgroup of \(H\), not necessarily \(H\): an isomorphism with \(H\) requires \(\varphi\) to be surjective.

The typical mistake is forgetting that the theorem relates the quotient to the image, not to the codomain. Another is believing \(G\) and \(H\) must have the same size; only \(|G|/|\ker\varphi|\) and \(|\operatorname{im}\varphi|\) agree. This result has versions for rings and modules, and it is the first of a family of isomorphism theorems that organize the structure of quotients.