How to visualize group elements as actions?

Question

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?

Answer 1

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 $\sigma_g(x)=g.x$ .