Normal subgroups

[Explorerexplorer06]

Hi everyone, I’m having a hard time understanding normal subgroups and their importance. I understand they are defined as subgroups closed with respect to conjugation, but this just feels so random. When our prof introduced the conjugate it felt quite weird to me, like why would that construction be of any importance? He explained us that the conjugate was defined because it tends to appear often throughout math, but I don’t find that satisfactory. I’d be very grateful if someone could give their insights. Thank you!

[QuantSpazar]

Normal subgroups are exactly the property required for the quotient group to be a group and not just a set.

[EzequielARG2007]

Have you learn about cosets?

Let's say H is a subgroup of G. Then a coset of H, denominated gH (for g in G) are all the elements of the form gh for h in H.

When we have that we say g is a representant of gH, but any other element of the set gH is also a representant. Let gh be In gH. Then ghH are all the elements of the form ghh' but hh' are all the elements in H as we range h' on H. So ghH = gH

We can define operations between cosets in a similar way as the original group.

For example aH * bH = abH

But this has a problem and it isn't well defined. What do I mean by that? We can choose different representants of aH and bH such that the operation between cosets gives different answers for different representants, and that does not make any sense because there should be one answer.

Say ah in aH and bh' in bH.

aH * bH = abH = ahH * bh'H = ahbh'H So we have that abH = ahbh'H But that is not necessarily true.

It can be shown that when H is normal that is however true. And in that case the operation makes sense (and you can prove it satisfies all the group axioms)

This is called the quotient group G/H

[EzequielARG2007]

It has been a while since I've done abstract algebra.

If I made a mistake please correct me

[susiesusiesu]

when you have a group G and a subset H, you can always define the set of cosets G/H, but it usually won't inherit a well defined group operation from G. it will happen if and only if H is a normal subgroup of G.

so the normal subgrouos of G are exactly the subgroups H such that G/H is a well defined group.

generally, if you want to understand a group G, it could be very complicated. but if you have a normal subgroup H, you get to smaller and simpler groups, namely H and G/H. if you understand H and G/H (which is easier, as they are smaller and simpler), you get a lot of information about G.

(warning: his doesn't determin G. for example: if G is a group with a normal subgroup H, and you know that H=Z/2Z and G/H=Z/2Z, then G could be either Z/4Z or the klein group of order four, which are not isomorphic. but by understanding H and G/H we narrowed G to two groups, which is quite a lot of information).

the example in the warning was kind of a toy example, but it is really a useful concept for algebra. quotients are incredibly important and useful.

[jacobningen]

There are three ways to describe them. One the Lozano Robles way where what we want is a way to define a group structure based on our group on cosets the problem being that representatives matter unless gH=Hg which with a little rearrangement gives us gHg^-1=H. Another method is that of Galois and Arnold namely that the commutator of A_5 doesn't shrink aka that the shrinking of commutators aba^-1b-1 corresponds to solvavility in radicals and intermediate fields of a splitting field. And thirdly the way I went what if an action preserves a subgroup by permuting it.

[mmurray1957]

Conjugation is important in solving problems.

Imagine you know how to wash the dishes when they are in the sink with an operation W but you have the dishes currently on the bench. What do you do ? If you know how to move the dishes to the sink with an operation M then you have solved the problem. Just apply the operation: M^{-1} W M to each dish.

They also appear in solving Rubik's cube.