What exactly is $\phi(G)$ in a homomorphism?

Question

I'm learning about group homomorphisms and I'm confused about what the $\phi$ transformation is exactly.

If we have some group homomorphism $\phi : G\rightarrow H$ what exactly does $\phi(G)$ mean?

Thanks for your time.

Answer 1

$\phi(G)$ is the image of $G$ under the mapping $\phi$, i.e. the set of all $\phi(x)$ for $x \in G$.

Answer 2

$\phi(G)$ denotes the image of the group $G$. This is actually a purely set-theoretic definition. Given a map of sets $f:X\to Y$, $f(X)$ denotes the image of $f$. That is, $f(X)\subseteq Y$ is $\{y\in Y: y=f(x)\:\text{for some}\:x\in X\}$.