Homomorphism Explorer

Let \(G\) be a group generated by a set \(S\). Any element \(g \in G\) can be written as a product of generators \(g = s_1 s_2 \dots s_k\). If \(\phi: G \to H\) is a homomorphism, then it must preserve this structure: \(\phi(g) = \phi(s_1)\phi(s_2)\dots\phi(s_k)\). Therefore, a homomorphism is completely determined by where it sends the generators of \(G\).

Von Dyck's Theorem (Extension Condition): A mapping of the generators of \(G\) extends to a well-defined homomorphism \(\phi: G \to H\) if and only if the chosen elements in \(H\) satisfy all the relations of \(G\).