I am not finding clear terminology in my abstract algebra book to be clear at least and my questions are simple. Consider the construction of a quotient group G': \begin{equation} G/K = G' \end{equation}
I know that $K$ can be shown to be isomorphic to the kernel of some homomorphism between $G$ and $G'$ (is there intuition behind this first homomorphism theorem?) Perhaps an example group would suffice. What is the proper terminology for $K$? Suppose $G, K$ and $G'$ were real numbers, then in arithmetic the terminology would be that $G$ is the divident, $K$ is the divisor and $G'$ is the quotient. What is the equivalent terminology in group theory?
Summarising:
- What is the intuition behind $K$ being isomorphic to some kernel of a homomorphism between $G$ and $G'$?
- What is the formal terminology for $K$?
Thanks for all the help!