Visual Group Theory, Lecture 5.3: Examples of group actions

[@algebraicgirl310]

Time-marks for anyone who needs them: 00:00 Overview 01:58 Groups acting on themselves by right-multiplication (Proof of Cayley's Theorem) 08:22 Groups acting on themselves by conjugation (The order of the Conjugacy class divides the order of G) 19:54 Groups acting on themselves by conjugation 31:35 Groups acting on cosets of H by right-multiplication

[@nm-de3bw]

On the proof of Clayley's theorem the kernels with different definitions are the same, but does this always happen? If so is there any proof?

[@nicolastxrluna]

congratulations for the excellent teaching greetings brazil!

[@fsaldan1]

Typo at 39'03'' should be x inverse on the right, not x.

[@kook6158]

Is there a typo at the last line of slide 5? Shouldn't it be r^(i+3)?

[@antoniolewis1016]

Yes you are correct

[@addemfrench]

Would you say that every lecture is equivalent to a single day of class? These lectures are truly great but making my head spin trying to digest them all this fast.

[@ProfessorMacauley]

In my experience, what I cover in a 50-minute class takes about 36 minutes in a KZbin lecture.

[@fsaldan1]

I don't think it is correct to say that pressing a button normalizes every subgroup if every subgroup is normal as in 31'. The subgroups are normal by hypothesis.

[@joetursi9573]

Hi, In the Stab(r^if), why is r^if listed twice?

[@chetedoherty]

I think that's a mistake, it should be r^(i+3)f

[@rasraster]

I had been very confused about the difference between stabilizer and kernel ... on first blush they seem to be essentially the same, yet stabilizers refer to a single element x and kernels to every element x. I think I have it figured out: the kernel is kernel of the *isomorphism*, and the image of isom is a set of permutations, NOT individual elements of S. So it's the identity *permutation* that is the image of everything in the kernel - and the identity permutation is the permutation that acts on every x and does nothing to it. Do I have this right? Thanks.

[@mistershoujo1235]

Stabilizer of a fixes a (but maybe other things as well!), Kernel fixes everything So the intersection of all of the stabilizers must be the kernel