# How to visualize group elements as actions?

I saw an excellent video of visualizing group elements as actions that transfroms one form of symmetry of an object to another.For example elements of $$(\mathbb Z,+)$$ can be viewed as shifting the real line grid(number line with all integers marked on it) by that integer.You see that two consecutive shifts yield the same effect as their additive effect would do,I mean they kind of gets added.Similarly elements of $$(\mathbb C,+)$$can be viewed as shifting the complex plane.Similarly the elements of multiplicative groups of $$\mathbb R$$ and $$\mathbb C$$ can be looked upon as magnifying,squishing,rotating the set.I liked this approach but I am not very expert in it.In general can I look upon any group element as some kind of action that I can easily visualize?Or it is just useful for non-complicated groups such as cyclic groups,abelian groups etc?Can someone help me to get a good grip of this approach by discussing some examples and suggesting some good text on such visual approach?

• I doubt every group can be meaningfully visualized this way, but some of the simpler and more familiar ones definitely can. For example, $S_n$ could be viewed as permuting the set $\{1,2,\cdots,n\}$ and the dihedral groups can be viewed as the various symmetries of a polygon (e.g. reflecting, rotating, which obviously have an associated action). I guess the issue is whether the group itself has a clear geometric or perhaps combinatoric basis. Oct 23 '19 at 3:26
• I would welcome visualizations of$\mathbb Z$,$\mathbb C$,$\mathbb R$,Circle group,$\mathbb R^*,\mathbb C^*$etc Oct 23 '19 at 3:29
• You can think of elements of a group $G$ as symmetries of the Cayley graph of $G$, though it should be noted that some symmetries of the Cayley graph of $G$ may not be in $G$ Oct 23 '19 at 6:57
• Can someone on provide me with visualization as group actions for standard group..? Oct 23 '19 at 7:26
• the complicated part of your question is "that I can easily visualize". Yes, any group can be seen as the group of symmetries of some "geometric" object. Now can you visualize it ? It will depend a lot on you, on what you call "visualize", and probably there will be examples where you can't, e.g. it's not clear how one might represent $\mathbb Z^{\mathbb N}$ Oct 23 '19 at 8:29

Study Dummit Foote and Judson's abstract algebra.Go through the chapter group action first,then study Cayley theorem.Cayley theorem tells us why we can think of an element of a group as a permutation,it will be more clear to you if you study group action where you will obviously encounter the following $$\sigma_g(x)=(g,x)$$ where $$()$$ denotes a group action.