# Definition A *[[group]] [[homomorphism]]* is like a [[Group Isomorphism|group isomorphism]] except it can be *many-to-one* and may only be *[[Types of Maps|into]]*. Like an isomorphism, we still have, for $g, h \in G_1$ and $f:G_1 \rightarrow G_2$, $ f(g \bullet h) = f(g) * f(h) $ In simpler language, the two groups are homomorphic if there exists a *many-to-one* correspondence between their [[Group Element|elements]]: $ \begin{align} a \rightarrow \hat{a},\\ b \rightarrow \hat{b},\\ ab \rightarrow \hat{a}\hat{b},\\ \end{align} $ where the "unhatted" letters are elements of the first group and the hatted ones are elements of the second group. If the correspondence is *one-to-one*, then it is an [[Group Homomorphism|isomorphism]]. # Properties Given a homomorphism $f:G_1 \rightarrow G_2$ with identities $e_1$ and $e_2$ respectively. Then: $ \begin{align} f(e_1e_1) &= f(e_1).f(e_1) = f(e_1) \\ &\implies f(e_1) = e_2,\\ f(g g^{-1}) &= f(g) f(g^{-1}) = f(e_1) = e_2 \\ &\implies f(g^{-1}) = f(g)^{-1} \quad \forall g \in G_1 \end{align} $