A Group Homomorphism is a structure-preserving map between two groups. Let $(G, \cdot)$ and $(H, *)$ be two groups. A function $\phi: G \to H$ is a homomorphism if for all $a, b$ in $G$: $$ \phi(a \cdot b) = \phi(a) * \phi(b) $$ In the context of cyclic groups $Z_n$ (integers modulo $n$) under addition, this condition simplifies to linearity modulo $m$: $\phi(a + b) = \phi(a) + \phi(b) \pmod m$.

Because cyclic groups are generated by a single element (usually 1), a homomorphism is completely determined by where it sends the generator. If $\phi(1) = k$, then $\phi(x) = \phi(1 + … + 1) = k + … + k = kx \pmod m$. However, not every $k$ works. The mapping must be well-defined, meaning if $x \equiv y \pmod n$, then $\phi(x) \equiv \phi(y) \pmod m$. This leads to the condition that $n \cdot k \equiv 0 \pmod m$.

An Isomorphism is a bijective (one-to-one and onto) homomorphism. If an isomorphism exists between two groups, they are structurally identical—they are essentially the same group just with different names for elements. For finite cyclic groups $Z_n$ and $Z_m$, they are isomorphic if and only if $n = m$.

The Kernel of a homomorphism is the set of elements in the domain that map to the identity (0) in the codomain. The Image is the set of all outputs in the codomain. The First Isomorphism Theorem states that the domain group modulo the kernel is isomorphic to the image.