Visual Group Theory, Lecture 3.7: Conjugacy classes

[@algebraicgirl310]

Timestamps who anyone who needs them: 00:00 Overview 03:15 Conjugacy classes 11:37 The class equation 15:17 More on conjugacy classes 20:35 Conjugacy classes in D6 example 28:32 Conjugacy preserves structure 33:19 Cycle type and conjugacy

[@addemfrench]

Man, I already knew the class equation and this visual presentation makes it make sense in a much more tactile way, very cool.

[@mistershoujo1235]

This is such a good series I love it

[@lancelofjohn6995]

Nice lecture, finally I understand the conjugation class of permutation cycle

[@GustavoPinho89]

Proszę bardzo, Bro!!! Thanks for coming back

[@thelastcipher9135]

The way the textbook used the word conjugate was really ambiguous to me. Thank you for this.

[@trailmixvideo]

At 10:03, I believe there is a typo, insofar as the last brace in the identity for D4 includes r rather than rf. A look at the visual partition should confirm this observation.

[@Mrpallekuling]

31:50: D5 has four conjugacy classes: {e}, {r, r^4}, {r^2, r^3} and {f, fr, fr^2, fr^3, fr^4}

[@astrophilip]

at 36:59, if you re-number 1 3, then (1 2) (2 3). so it's easier to just conjugate g or h by (1 3) to get the other.

[@hanyanglee9018]

18:23 Does anything show that the orders of x and y could be different? If the conception of order is defined with the minimal times an element needs to multiply itself to get back into the identity element, x to the n ==e, means the n is already the smallest due to the definition. But it's ok, it doesn't do anything to the conclusion.

[@Elgeneralsimo69]

The conjugation relation at 39:00 _seems_ related to the naturality square or the adjunction triangle identities of Category Theory. Thoughts? www.euclideanspace.com/maths/discrete/category/comparing/adjunction/index.htm

[@professorno9080]

great video! thanks

[@sahhaf1234]

I think there is a mistake @0:30 : It is written on the slide "....for H

[@goblin5003]

If H is a subgroup of G, then the conjugate of H by a fixed g in G is indeed a subgroup of G However this conjugate of H by a fixed g in G is not a subgroup of H, if that’s what you meant

[@sahhaf1234]

@@goblin5003 Tess, that's exactly what I've meant. He had to specify that H is a subgroup.. I have first encountered the definition of conjugacy in conjugate sets. So I rarely assume that what is conjugated is a group. Thanks,

[@sajateacher]

24:04 - 120 and 240 degrees?

[@Mrpallekuling]

D6 is a hexagon, so r = 60 degrees, r^2 = 120 degrees, and r^4 = 240 degrees

[@sara-ql1xs]

Hey, can someone help me please? Why is it that at 17:35 we can conclude that. |x| >= |gxg^-1|? Wouldnt it be the other way around? Sorry if I sound dumb😂 amazing video btw😍

[@sgut1947]

I agree it isn't very clear. Let n be the _smallest_ positive number such that x^n=e, so that n = |x|. Then we know that y^n=e because of the way (gng^(-1))^n collapses. But we don't yet know that y^m is not equal to e for some m

[@hanyanglee9018]

what is the difference between center and normal subgroup? Center is biggest among all the normal subgroups?

[@Mrpallekuling]

The center is the set of elements that are (as individual elements) invariant under conjugation by other elements. But a normal subgroup must be invariant under conjugation as a *set* (conjugation may move elements within the set).

[@hanyanglee9018]

@@Mrpallekuling Thanks. I guess I would come back later since this is a bit too hard for me.

[@CaptchaSamurai]

Superb! Thank you ε>

[@Nmmoinn]

28:23 "2+2+2+3+3=6" I know you meant 12 but just found this a lil funny