Quotient of a group by kernel of group homomorphism

Question

So I was presented with the following definition and proposition:

Let $(A, +)$ and $(B, +')$ be two abelian groups and let $f: A \rightarrow B$ be a morphism. We define the kernel of a morphism as $$ker(f) = \{a \in A : f(a) = 0_B\}$$ where $0_B$ denotes the neutral element of $B$. Then $ker(f)$ is a subgroup of $A$ and there is a unique morphism: $$\bar{f}: A/ker(f) \rightarrow B$$

This is the first time I encountered the notion of a kernel, but I think I understand the concept. I also know about quotient groups, I get the definition of splitting up a group by an equivalence relation into equivalence classes and I also understand the common example $\Bbb{Z}/5\Bbb{Z}$ and similar.

However, I can't wrap my head around what $A/ker(f)$ really is, or how to visualize it in my head. Like, if for example $ker(f) = \{0_A\}$ then $A/ker(f)$ is just $A$. And if $ker(f) = A$, then $A/ker(f)$ is $\{0_A\}$, right? That is simple enough and makes sense to me. However, what if $ker(f) \neq A$ and has more than one element?

Answer 1

"However, I can't wrap my head around what A/ker(f) really is, or how to visualize it in my head."

Its a quotient group $G/N = \{x+N\mid x\in G\}$ where $G$ is a group and $N$ is a normal subgroup.

For instance, take $G=\Bbb R^n$ and $U$ be a subspace. Then $G/U$ consists of the affine subspaces $x+U$ which are parallel to $U$ with "shift vector" $x\in G$.

The example $f:\Bbb Z\rightarrow\Bbb Z$ is a bit misleading, since the only homomorphisms are the zero mapping and the identity mapping.

Answer 2

Wuestenfux already gave you a set theoretic description of the quotient group. But on a conceptual level, you should view a quotient group as the image of a group homomorphism with a specific kernel. Specifically, you should conceptualize the group $G/\ker\varphi$ As some group which behaves exactly as the image of $\varphi$, up to relabeling. This idea is made rigorous by the fundamental theorem on homomorphisms, which has a corrolary that says $G/\ker\varphi\cong\operatorname{im}\varphi$ for any group $G$ and any group homomorphism $\varphi$ with domain $G$.

The set theoretic definition given by Wuestenfux is then a specific incarnation of such a quotient group, which can be used to show that such a quotient group exists in the first place.