I noticed I was confused and liable to forget my grasp on what the hell is so “normal” about normal subgroups. You know what that means—colorful picture time!
First, the classic definition. A subgroup H is normal when, for all group elements g, gH=Hg (this is trivially true for all subgroups of abelian groups).
ETA: I drew the bounds a bit incorrectly; g is most certainly within the left coset (ge=g).
Notice that nontrivial cosets aren’t subgroups, because they don’t have the identity e.
This “normal” thing matters because sometimes we want to highlight regularities in the group by taking a quotient. Taking an example from the excellent Visual Group Theory, the integers Z have a quotient group Z/12 consisting of the congruence classes ¯0,…,¯11, each integer slotted into a class according to its value mod 12. We’re taking a quotient with the cyclic subgroup ⟨12⟩.
So, what can go wrong? Well, if the subgroup isn’t normal, strange things can happen when you try to take a quotient.
Here’s what’s happening:
Normality means that when you form the new Cayley diagram, the arrows behave properly. You’re at the origin, e. You travel to Hg using g. What we need for this diagram to make sense is that if you follow any h you please, applying g−1 means you go back toH. In other words, ghg−1=h′∈H. In other words, gh=h′g. In other other words (and using a few properties of groups), gH=Hg.
I noticed I was confused and liable to forget my grasp on what the hell is so “normal” about normal subgroups. You know what that means—colorful picture time!
First, the classic definition. A subgroup H is normal when, for all group elements g, gH=Hg (this is trivially true for all subgroups of abelian groups).
ETA: I drew the bounds a bit incorrectly; g is most certainly within the left coset (ge=g).
Notice that nontrivial cosets aren’t subgroups, because they don’t have the identity e.
This “normal” thing matters because sometimes we want to highlight regularities in the group by taking a quotient. Taking an example from the excellent Visual Group Theory, the integers Z have a quotient group Z/12 consisting of the congruence classes ¯0,…,¯11, each integer slotted into a class according to its value mod 12. We’re taking a quotient with the cyclic subgroup ⟨12⟩.
So, what can go wrong? Well, if the subgroup isn’t normal, strange things can happen when you try to take a quotient.
Here’s what’s happening:
Normality means that when you form the new Cayley diagram, the arrows behave properly. You’re at the origin, e. You travel to Hg using g. What we need for this diagram to make sense is that if you follow any h you please, applying g−1 means you go back to H. In other words, ghg−1=h′∈H. In other words, gh=h′g. In other other words (and using a few properties of groups), gH=Hg.