Visual Group Theory, Lecture 2.1: Cyclic and abelian groups

Visual Group Theory, Lecture 2.1: Cyclic and abelian groups
In this lecture, we introduce two important families of groups: (1) "cyclic groups", which are those that can be generated by a single element, and (2) "abelian groups", which are those for which multiplication commutes. Additionally, in any group, every element generates a cyclic (sub)group called its "orbit". We can visualize the orbit structure of a group by an object we call a "cycle graph".

[@MrBebopbob]

Absolutely wonderful!!! This clarifies information about group theory in a way I have not seen by looking at many textbooks and technical papers. Thank you for posting these lectures on line. I look forward to viewing (and maybe re-viewing) these lectures.

[@Anonymous-pm7qc]

It looks like theres a mistake at 25:01. the right bottom writing should say = {e,r^2} instead of = {e,r}

[@FrenzyLi]

Reminder for future visitors: the "orbits" in this specific lecture refer to the subgroups (of D_3) generated by each of its (6) elements. The actual orbit is commonly used to refer to the movement of elements under group action.

[@rubberubertuber]

I agree. The video is a bit confusing if you're already vaguely familiar with the other use of "orbit".

[@anfarahat]

Yes, just had a comment about that. The use of orbit is confusing.

[@ReidarWasenius]

Excellent material!! Thanks for producing and posting it. :-)

[@lindseystevenson1253]

The peppers have 10 "arms" not eight, so the rotation should be 36° rotation

[@anfarahat]

Thank you for the great explanation and presentation. It is one of the best.
I am confused by the definition of an orbit you provide. My understanding is that an orbit of an element s in a set S is the set of elements G sends s to through its group action on S.
How does that generalize to G acting on G? Do we just consider the orbit of the subgroup formed by one generator of G? In that case, would the orbit of an element g' in G be the right coset formed by a the subgroup generated by some element f?

[@ucojq]

Is this use of the term "orbit" standard/typical? I'm used to defining the orbit of an element x in group G the following:
G(x) = {gx | g in G}.
I tried to find this use of orbit elsewhere, but at least Mathworld and Wikipedia seem to agree with the definition I'm used to.

[@MsSlash89]

The two definitions have slightly different connotations. The notion of orbit the video described is used in the context of group generators, cyclic groups and subgroups. The definition you have given is about group per se, but it’s about group actions. If I have read correctly and understood what you are talking about, x need not to be an element of the group. You can think of x like some part of a physical object, like one vertex of a square. You then apply a group to the object, for instance, the Cyclic Group C4 would turn the square. Hence, the orbit of x is the set of all new positions the vertex x can be brought to by applying the actions in the group C4. We say that the group C4 acts on the square. In this case, the orbit of x is all four vertices.

[@anfarahat]

Yes, that's what I am used to. In the case you are mentioning, if x is an element of G, Gx = G, which is not very useful. Yet, considering subgroups of G will definitely make more sense here. In the example above, I believe the subgroups considered are those generated by recursive application of every element of the group to itself, generating cycles.

[@shocker02k]

This video made me finally understand e. Thank you!

[@jameyatesmauriat6116]

Can I know how did you understand the Cayley diagram because it’s confusing to me?

[@scitwi9164]

22:25 Hmm... but in a group, every node must have an arrow `g` going out to some other node, right? Because if it hadn't, we could get stuck in some node, being unable to apply certain operations (`g` in this case) while at some "dead end" nodes. But the group axioms require us to be able to perform every group operation always, in whatever order we like. So this implies that if we follow the arrows of `g` (and we can always do that), we must at some point return to where we started (if the number of steps was finite), or never come there (in that case the number of steps is infinite). So in a finite group there will always be closed orbits, for every generator. Am I right? If that is the case, then cyclic groups are really the building blocks of all other groups :>

[@rish5827]

Well groups are built from generators. If the group is of finite order the generators must be of finite order. Other than the number of generators and their orders, groups are built from relations. So you can think of combining the generator orbits with certain relations to build finite groups.

Commuting diagrams is useful terminology if you're using this as a jumping off point into category theory.

[@xamogxusx]

So so great!

[@ijindela5722]

2:20. - There are 10 peppers, not 8...

[@jameyatesmauriat6116]

I don’t understand how the Cayley diagram works , how have you obtained elements and orbits from the diagram?Anyone please clarify this to the greatest extent?

[@msadeeq6484]

from where can i find these lectures slides please

[@Harsimran_Singh_27]

Where can we find these slides?

[@maurocruz1824]

15:30

[@blomman43]

His name is Niels Abel.