Visual Group Theory, Lecture 3.6: Normalizers
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- Опубліковано 7 сер 2024
- Visual Group Theory, Lecture 3.6: Normalizers
A subgroup H of G is normal if xH=Hx for all x in G. If H is not normal, then the normalizer is the set of elements for which xH=Hx. Obviously, the normalizer has to be at least H and at most G, and so in some sense, this is measuring "how close H is to being normal". We interpret this in terms of Cayley diagrams, and then prove some basic properties of normalizers: they are always subgroups, and they are unions of cosets -- precisely, those left cosets that are also right cosets.
Course webpage (with lecture notes, HW, etc.): www.math.clemson.edu/~macaule/...
Motivating the normalizer by quantifying the notion of normality through voting is a great pedagogical start.
this lecture series is SAVING me!!! thank you so much for uploading these!
Greetings from Brazil. This series is my favorite soap opera! :)
I love the opening question.
You are awesome Professor! Even though my semester is over I am binge watching your videos instead of Netflix! Thanks a lot and looking forward to more amazing math! Is it possible for you to do a lecture series on representation theory, I am gonna take that course next sem and I am a final year undergrad student from India! Thanks a lot!
Very good explanation!
Voting idea for explaining is perfect, thank you professor 🙏😊😃
In every math course there's a point in which my neuron snap.
no wait...
yeah it's gone.
13:10
I have a problem with the proof of observation 1. Why is the last equality (i. e. Hg=Hb) true? We only assumed that gH=bH, in other words b \in gH. For this equality we would need b \in Hg.
Ah I see, that's because gH=Hg so b\in gH is equivalent to b \in Hg. Ok
I guess the color of x at 8:59 should be blue..
Yes, same error as in the previous lecture (3.5).
Gerrymandering of group theory
1:30 Should be "at minimum ONE element (e) votes "yes""
No, this is correct. Notice every g in H will always satisfy gH=Hg because H is a subgroup.