# Definition: Homomorphism (group)

Definition 30.1 Let $G$ and $H$ be groups, and let $f$ be a function from $G$ to $H$, i.e. $f:G\rightarrow H$.  If $f$ has the special property that, for all $a,b\in G$ it is true that $f(a\cdot b)=f(a)\cdot f(b)$, then $f$ is called a homomorphism.

This definition is extremely important, and should be thought of as a natural way of “specializing” a function.  Namely, we can define a function from any set to any set, and therefore for any two groups we can define a function from one group to the other (since groups are just special types of sets).  Once we choose two groups, there are many (sometimes infinitely many) different functions from one to the other that we can define.  However, since we’re dealing with groups (as opposed to just generic sets), there is more mathematical structure “floating around”, and homomorphisms are the subclass of functions that interact with this extra structure in a particular and natural way.  To read more about homomorphisms, check out lesson 30 where we introduce and discuss them in more depth.

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