Definition: Homomorphism
A homomorphism is a total function \(f:G\to H\) between two algebraic structures $(G,\ast)$ and $(H,\cdot)$, which for all $x,y\in G$ fulfills the property
\[f(x\ast y)=f(x)\cdot f(y).\]
Notes
- Homomorphisms can exist between different types of algebraic structures, including groups, rings, and fields.
- A homomorphism means that it doesn't matter if we first combine the two elements $x,y$ via the operation "$\ast$" in $G$ and then map the result to $H$ or if we first map $x$ and $y$ to $H$ and combine $f(x)$ and $f(y)$ via the operation "$\cdot$" in $H$. In both cases, we will get the same result for all pairs of elements $x,y\in G.$
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