The quotient group is the result of a simplification done by an homomorphism
https://math.stackexchange.com/a/69063/53203 mentions that the quotient subgroup is a type of subgroup but "with less information".
This is also where my intuition lies, but I would like to make that a bit more precise by uttering the key missing keyword: "homomorphism". This is going to be a more focused subset of this other answer: Intuition behind normal subgroups
An isomorphism is a bijective function between two groups (of the same size since it's a bijection) and means that they are the exact same as far as the group structure is concerned. Pretty boring.
An homomorphism however does not have to be a bijection, only surjection: it can take a larger group and transform it into a smaller image group. Notably, several distinct inputs can map to the same output.
The tradeoff is that this smaller group contains a "coarser" group structure than the original group, as it ignores some finer part of the original group (preview: that finer part is the normal subgroup structure). This image structure is simpler because the homomorphism can map multiple input elements to a single output element.
Now, as I have explained in more detail at: Intuition behind normal subgroups:
by the fundamental theorem on homomorphisms, there is a one to one relation between homomorphisms quotient groups (or normal groups):
- for every homomorphism, the image is a quotient group
- for every quotient group, there is a corresponding homomorphism
Therefore, the quotient group is always the result of a simplification done by an homomorhpism.
I like this intuition, because it is very easy to understand what an homomorphism is: it is just a function that keeps group structure.
And now we've just seen that every quotient group maps one to one to this thing that is easy to understand.
By the fundamental theorem on homomorphisms, this is how every homomorphism looks like.
From this it is clear how the structure of the quotient G/N is simpler than the original G: we collapsed the structure of the entire normal group N to a single point! Therefore, an homomorphism is basically a simplification function that ignores the structure of the normal group while doing its transformation.
This also makes it clear why the name "quotient" is used: it is because we are splitting the larger group G into two smaller groups:
- the normal group N
- the quotient group G/N, which is specified by N
which gives us a clear analogy to integer multiplication and division.
It is important to note however that this intuition is only valid in the "division sense": because groups multiplication is more complex than integer multiplication (notably, non-abelian), there is no simple known way of "multiplying two smaller groups to recover a larger group".