Question Regarding Equivalence of Definitions of Homomorphisms of Groups

Question

I'm new to group theory and was wondering what a "map" means in this context. I was also hoping someone could explicitly state what the proof was conveying in words rather than symbols. Thank you.

Answer 1

In this context a map is just a function. If you're trying to understand what an homomorphism is, maybe you can try an example. By definition, if $(G, *), (H, \cdot)$ are groups and $\phi:(G, *)\to (H, \cdot)$ is a map, then $\phi$ is said to be an homomorphism if it satisfies the property $$ \phi(a * b)= \phi(a) \cdot \phi(b)$$ for all $a,b \in G.$ Consider, for example, the real numbers with the addittion $(\mathbf{R},+)$ and the nonzero real numbers with the product $(\mathbf{R}-\{0\}, \cdot).$ These are two (different) groups. Define $\phi:(\mathbf{R},+)\to (\mathbf{R}-\{0\}, \cdot)$ by $$ \phi(x)=e^x.$$ Then, if $a,b \in (\mathbf{R},+),$ we have $$ \phi(a+b)=e^{a+b}=e^a \cdot e^b = \phi(a) \cdot \phi(b),$$ so $\phi$ is an homomorphism of $(\mathbf{R},+)$ into $(\mathbf{R}-\{0\}, \cdot).$