Question

Why do we take normal subgroups when we define a quotient group?

Answer 1

et’s pretend we didn’t know that. Let G be a group and H a not-necessarily-normal subgroup. We want to define G/H . To me, the most intuitive characterization of G/H is that you don’t care about “multiples of stuff in H .” This is borrowed straight from the concept of modular arithmetic, so it’s going to be my guiding principle in trying to define G/H .

 

The other guiding principle is that, since we’re in group theory land, I also want G/H to be a group. (After all, the integers mod 5 isn’t very useful if you couldn’t do arithmetic mod 5.)

 

Okay, let’s proceed normally: define G/H to be the left cosets of H ; that is, {gH} for g∈G . We want the natural group operation: if g1H and g2H are cosets, the natural operation is defined by (g1H)(g2H)=g1g2H .

 

Looks right, but it actually doesn’t make sense. Why? Because there’s more than one way to represent g1H , at least if H is not just the zero subgroup. In more detail, it might be that f1H=g1H , with f1≠g1 . By way of example in modular arithmetic, the integers 1, 4, and 10 “represent” the same number mod 3. In the case of modular arithmetic, we have the benefit of the integers being ordered, so we can all agree to pick a standard representative: the one between 0 and n in the integers modulo n . But with a generic subgroup H of a generic subgroup G , there’s nothing special going on to help us pick a standard representative.

Because otherwise it doesn’t work.