GT15. Group Actions

Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group.

[fionachristie]

If I could come and give you a hug I would, my lecturer seems to like to go out of his way to make things difficult- this makes MUCH more sense. Thank you!

[MathDoctorBob]

You're welcome! Personally I use the group acting on itself often, and there are several ways to do this. Giving it a name just clarifies which one I'm using. When no chance of confusion, I drop it.

[MathDoctorBob]

Your welcome! A better thing to show would be: if x and y are in the same orbit (so gx=y for some g), then Stab x and Stab y are conjugate.

You probably need a little more. Stabilizers can be normal: G abelian, action by conjugation, all stabilizers are G itself (normal). Of course, each point is an orbit here, so you may want a closer look at conditions like faithful, transitive, simply transitive, etc.

[MathDoctorBob]

It will depend on the example. Three good examples to know: symmetric/alternating groups, dihedral groups, and matrix groups.

Matrix groups: if small, say 2x2, solve AX=XA instead of AXA^-1 =X.

Symmetric/alternating groups: rule for conjugation as relabeling a permutation

Dihedral groups: use presentation c^2=e, r^n=e, crc^-1 = r^-1.

[MathDoctorBob]

Say the elements of D8 are of the form r^k and cr^k, where c is the reflection in the x-axis and k is between 0 and 7. c and r satisfy the relation crc^-1 = r^-1. (Since c^2=e, c=c^-1.)

First note that all r^k centralize r. That must be it. This subgroup has index 2, so if any larger, it would be all of D8. But crc^-1 = r^-1, so c is not in the centralizer. Thus the conjugacy class has order 2, and we see immediately the elements are r itself and crc = r^-1.

[MathDoctorBob]

You're welcome!

Transitive - given points x and y, we can always find a g with g.x=y. That is, we can move between any two points using the group action. Example: R acts on itself by addition r.x = r+x. Not transitive: R\0 acts on R by multiplication.

Faithful: the only solution to g.x=x for all x is g=e. Example: R acts on itself by addition. Not faithful: R\0 acts on R by r.x = |r|.x

Regular: there exists a unique g such that g.x = y. Example: R acts on itself by addition

[Wolfgang]

Orbit-Stabilizer theorem just clicked and made sense. Thanks so much Dr!

[MathDoctorBob]

It might also help to review GT6. on centralizers and normalizers. These are stabilizers of conjugation on elements and subgroups, respectively.

[MathDoctorBob]

Do you have something specific in mind? Here's a good exercise. When n is 5 or larger, Sn (symmetric group on n letters) has only {e}, An and Sn as normal subgroups. But generating stabilizers subgroups is natural here. Consider the natural action of Sn on X1={1, ..., n}. If S is a subset of X1 with m elements, then Stab S is iso to Sn-m, usually not normal. More general let the points of X2 be subsets with k elements (so subset of the power set of X1). Pick m subsets, and try small n.

[Tehom]

19:52 It is not necessary to correct yourself here. Two reflections generates a rotation, so by including all the reflections you already included all the rotations.

[MaryUkraine]

Thank you soooo much for explaining all the definitions in English!!!

[MathDoctorBob]

@The1000ankit What do you have in mind specifically? - Bob

[MathDoctorBob]

@wdlang06 Thanks! I do representation theory, so I'm always trying to learn more physics. - Bob

[cpaniaguam]

In a past video (Example of group action, I think), I commented on the notion of when actions are faithful. Here we talk about the possibility of actions being transitive, which is the same as surjective or onto. Why, in the realm of group actions, these notions have different names? Is there some historic reason?

[memequbbaj]

Thank u dr for your excellent work, but I wonder are there more examples on : when the stabilizer is not normal ?

[wdlang]

you teach very well. i study theoretical physics, i like your lecture.

[accidentproneathlete]

Thank you for the excellent videos - they're an excellent reference tool to help me with my class notes. One question however: what is the advantage of defining a group action as (g, x) -> pi(g) x
as opposed to (g, x) -> g x
as my lecturer decided to use?

[memequbbaj]

thank you dr, I was trying to prove that stab(x) is not a normal subgroup .

[andreapotylycki]

thank you so much. this is sooo helpful!

[sacinesco]

When you say the permutation acts on "i", are you saying you permute just one elment in the set X, or permute all elements i in X?