Kernel, Sets and Logic

Question

Given the groups G, H the kernel of a homomorphism $f : G \rightarrow H$ is defined as : $\{$ $g \in G$ : $f(g)=e_H$ $\}$. I was wondering, is there a way to express the kernel in terms of quantifiers or/and if and only if statements? Moreover can the image of the homomorphism also be expressed in terms of quantifiers or/and implications. What I mean is, are there logically equivalent expressions other than the above expression?

Answer 1

Sure. Here is a few:

• Given $h \in H$, $h$ is in the image of $f$ if and only if $\exists g \in G$ with $f(g) = h$.
• Given $g \in G$, $g$ is in the kernel of $f$ if and only if $\forall x \in G$, $f(x) = f(gx) = f(xg)$.
• Given $g\in G$, $g$ is in the kernel of $f$ if and only if $\exists x \in G$ with $f(xg) = f(x)$.