301.9C Extra: Normal Subgroups and Conjugacy Classes

A normal subgroup is exactly a subgroup obtained from a union of conjugacy classes. This observation goes a long way to (1) explaining why the cosets of a normal subgroup can behave like single elements, and (2) determining what normal sugroups a given group may have — example S4 in this video.

[binaykumbhar]

1+3+8=12 (i.e. N=subgroup formed from Cycle type 1+1+1+1 & 2+2 & 3+1) which devides 24 also. Clearly it's a subgroup of S4 too.
And i checked for all a belongs S4 and n belongs to N,
ana^- also belongs to the N.


Why can't it be a Normal to S4 then ?

[hyperduality]

Subgroups are dual to subfields -- the Galois correspondence.
"Always two there are" -- Yoda.
Even (symmetry, Bosons) is dual to odd (anti-symmetry, Fermions) -- wave/particle duality!

[binaykumbhar]

I have a doubt .

Why did you take union of two conjugacy classes not more ?


Is a normal subgroup is a union of only two conjugacy classes ?

That's because 1+3+8=12/Order of S4 (i.e. Subgroup formed from union of 1+1+1+1 ; 2+2 & 3+1 cycle type) but you took only 1+3=4/24.

[PunmasterSTP]

I'm not immediately seeing why in S4 conjugate elements have to have the same cycle type, but it'll probably occur to me randomly sometime...

[maurocruz]

"what makes 
a subgroup a normal subgroup 
is our inability to tell the differences 
between different elements in its same Coset"

[CraaaabPeople]

I don't know if it is possible, but would you be able to discuss what a normalizer of a subgroup is? I know these videos are part of a course, but just a request!