The First Isomorphism Theorem

In the setting of Groups: Standing Notation, let $G$ and $H$ be groups and $f:G\to H$ a group homomorphism, with kernel $K$ and image $f(G)$. Here $K$ is a normal subgroup of $G$ by Kernel and Image of a Group Homomorphism are Subgroups (iii), so the quotient group $G/K$ is a group by The Product of Cosets of a Normal Subgroup Is Well Defined and Makes the Cosets a Group (ii), and $f(G)$ is a subgroup of $H$ by Kernel and Image of a Group Homomorphism are Subgroups (ii), a group under the restricted operation by Subgroup Criterion and Basic Examples (ii).

(i) The induced isomorphism. There is exactly one function $\bar f:G/K\to f(G)$ such that

$$\bar f(gK)=f(g)\qquad\text{for every } g\in G,$$

and $\bar f$ is a group isomorphism from $G/K$ onto $f(G)$.