🤨 A group theory problem utilizing centralizer and normalizer

A little group theory problem! :-D Share your thoughts in the comments below!

Outro:
Lateralus, as performed by Sakis Strigas
Originally by Tool

My current equipment for making videos (affiliate links):

[EpicMathTime]

Error: at 4:17, this should be another inequality, not equality. As mentioned, (|N|-1)! is an *upper bound* on the order of Aut(N).

Thank you Haru Shaito and Viktor Kronvall for pointing this out.

[cezarstroescu]

I didnt understand anything but watched to support you

[ArtArtisian]

Ha. This is the first time I actually wanted a SES picture proof. S_n-1 acts on N by permutation, map G into S_n-1 according to their action on N by conjugation. This is a group hom (function restriction), with kernel equal to the centralizer. Finish with
|G| leq |im| |ker|
The map from G is not necc injective, but restricting S_n-1 so that it is only improves the bound. Replace S with aut if you prefer.

[Jotakumon]

I think the solution to the last part is N = {(1), (12), (34), (12)(34)} (and G = S_4).

I think I got to the point where I proved that this is the only normal subgroup with 4 elements of G with 4 elements, thus it should be the only candidate. My issue now is proving that C_G(N) <= N.

C_G(N) is a subgroup of G, so if C_G(N) is a subset of N, it should also be a subgroup of N. There are only two subgroups of N, which are {(1)} and N itself. But I don't know how to determine if it's either without brute-force testing it. So I think a way to go would be to take any x in C_G(N) and prove that this needs to be in N, thus C_G(N) is a subset of N, which would be enough. But I can't get past that. And that is provided that the rest is correct.

Any helpers out there?

EDIT: It's not the only solution. Permuting the numbers gives more, so {(1), (13), (24), (13)(24)} and {(1), (14), (23), (14)(23)} should also be solutions.

[gdsfish]

If I pass my upcoming calc 2 exam, I might be able to come back to this video half a year later and be like: "Oh... so that's what he was talking about..."

[umbraemilitos]

I was thinking about normalizers and centralizers recently.

[COZYTW]

>never studied group theory beyond simple sets & subsets
>doesn't know what centralisers and normalisers are
>commented to help EMT's Youtube algorithm


Hopefully I can one day reach that realm of Maths where I can solve proof-by-concept questions for breakfast

[alkankondo]

Nice! Well-edited video with clear explanations!

[duncanw]

I'm really happy about this; theres not enough good, concise abstract algebra content on here. You and fematika are about all.

[IbrAhMath]

A naive question: In 05:45 why is the order of the automorphism group of N in the last line exactly equal to (|N|-1)! while in the first line we can also have inequality?

[estebanmartinez]

Nicr video!
By a similar argument and applying Lagrange's theorem (the order of a subgroup of a finite group divides the order of the group) it can be shown that in fact |G| divides |N|! wich implies the result of the video

[Grassmpl]

I'm having trouble proving groups of order 540 is not simple. Any advice or sources? I know how to count the Sylow p-subgroups for p=3, 5. I just want to find a proper subgroup of index up to 8 and I'm done since 540 is not a factor of 8!, but is factor of 9!.

[duncanw]

When the yt autocaption says "by jek shion's"

*HMM*

[alvinlepik]

The exercise at the end is equivalent to finding a normal subgroup of S4 of order 4.

[chemistro]

where are all those internet math gods that were arguing in the comments about 6/2(1+2) and 0.999....? Surely they can give some great insights on this problem as well? 😂

[neptunian]

What happened to the other group theory video where you were doing homework and worked out some characteristic table?

[moskthinks]

6:10 immediately thought of the Klein Four Group, which I know is normal.

[HotPepperLala]

I solved it using the hints, was this problem from a book and the hints were from a lemma or something? I initially tried it without the hints and I think this can be broken down with Lagrange's theorem by cases.

[morgengabe]

Why do automotphisms need to fix the identity element?

[duncanw]

Bro is it the convention to use > for subgroup? I don't like that.