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For a group homomorphism $\phi :G\to H$, the group isomorphism $g\ker \phi \mapsto \phi (g)$ givesThe group kernel records which inputs become indistinguishable, and the image records which outputs actually occur. The first isomorphism theorem says that identifying precisely those indistinguishable inputs gives a group isomorphic to the image. It is one of the general isomorphism theorems. ^[ref].

For example, reduction modulo three gives a surjective group homomorphism from the additive cyclic group $\mathbb{Z}/6\mathbb{Z}$ to $\mathbb{Z}/3\mathbb{Z}$. Its group kernel is $\{0,3\}$. The three cosets $\{0,3\}$, $\{1,4\}$, and $\{2,5\}$ become the three outputs zero, one, and two, respectively. ThusThe quotient group diagram shows both the identification and the resulting operation. More generally, fibres of a group homomorphism are cosets of its kernel explains why this map is well-defined and injective; it is surjective onto the image by definition.

The isomorphism theorems relate homomorphisms, their kernels and images, and quotient constructions in algebra. For groups, the first isomorphism theorem identifies a homomorphism's image with the quotient group by its group kernel. Analogous statements include the first isomorphism theorem for rings and the First isomorphism theorem for vector spaces: identifying precisely the inputs with the same output preserves the remaining algebraic structure.