What does it mean by killing a subgroup in quotient process?

Question

We often use the word killing $H$ while describing the quotient $G/H$,where $H$ is a normal subgroup of $G$.In this process what we actually do is we consider all the cosets induced by $H$ and consider them as single point,i.e. squish the whole cluster to a single point.What is the meaning of the work kill in this context?I want to understand the precise definition of kill.

Answer 1

It can be useful to think of a quotient group in terms of a group homomorphism $\pi : G \to G/H$, where $\pi(g) = gH$. Then the quotient “kills off” $H$ in the sense that $\ker \pi = H$.

Answer 2

Well, it means that every element of $H$ is sent to the neutral element $e$ of $G/H$. As $e$ is trivial in many if not all contexts, the peculiarities of $H$ do not survive the construction of the quotient group and are therefore killed.

Answer 3

If you have seen quotient spaces in topology, then this corresponds to the idea of collapsing a subspace to a point. What you are doing here is that you take a subgroup, and then you identify each element in this subgroup with the identity element. You can show that cosets of a subgroup partition any group, so this will give you an equivalence relation, and the equivalence classes will be the cosets of your subgroup, the internal structure of which you have lost in this process. As everything in the subgroup is identified with the identity element, if the subgroup is normal it becomes the identity element of the quotient group. You have trivialised the whole thing, you have killed it.

If you want a simple example, consider the integers. You may or may not be surprised to find that $1\neq2\neq4$, but if you then consider the quotient $\mathbb{Z}/2\mathbb{Z}$, you will notice that $1$ and $2$ correspond to different different elements in the quotient, but $2$ and $4$ have been identified. You have lost the ability to distinguish between different even (or odd as well) integers. Dead men tell no tales.