After defining a homomorphism of groups as an operation-preserving function, we now ask what the condition forces automatically. A homomorphism does not merely preserve one chosen product; it controls identities, inverses, powers, subgroups, normal subgroup preimages, commutativity inside the image, and orders of elements. The focus keyword basic properties of homomorphisms points to these unavoidable consequences. Students often try to verify each consequence separately from the beginning, but most of them follow from the single formula $f(ab)=f(a)f(b)$. In this lesson we prove the standard properties that will later make kernels, quotient maps, and isomorphism tests work efficiently.

Let $G$ and $G_1$ be groups with identity elements $e$ and $e_1$, respectively. If $f:G\to G_1$ is a homomorphism, then $f(e)=e_1$ and $$\begin{equation*} f(a^{-1})=f(a)^{-1} \end{equation*}$$ for all $a\in G$.