Intuition behind group action on a set

Question

In Algebra Chapter 0 the definition of a group action on a set is given as: An action of a group $G$ on a set $A$ is a set function $P:G\times A\rightarrow{A}$ such that $P(e_G,a)=a$ and $P(gh,a)=P(g,P(h,a))$ $\forall g,h \in G\,\forall a\in A$ the first condition makes sense but I am unsure of the purpose of the second condition, it seems rather strange. My thinking is it somehow preserves some aspect of $G$ but I am unsure. If someone could explain the reasoning or intuition behind the 2nd requirement that would be great.

Answer 1

I assume your second condition is meant to say $P(g,P(h,a))=P(gh,a)$. The purpose of this is to say that the group action is "associative" in some sense. It is far easier to see if we write $g\cdot a$ for $P(g,a)$. Then the condtions become $e_G\cdot a=a$ and $g\cdot(h\cdot a)=(gh)\cdot a$ for all $g,h\in G$ and $a\in A$. Hopefully this makes the intuition clearer.

Answer 2

My philosophy is that groups are to group actions as potential energy is to kinetic energy. You might recall reading that originally, historically groups were first conceived of as sets of functions that acted on some thingie (points in space, numbers in a number system, whatever). The more pointed term for this is that a group is a set of symmetries. As such, whenever an object exhibits symmetry, in other words if there are transformations that preserve it, then these transformations can be composed and inverted. That's where the idea of a group comes from. This idea was then abstracted into just a set with an associative binary operation having inverses and identity.

Group actions recover the "function" nature of group elements; they get to go hunting out in the world and do things to other things. They act as functions on some set, or symmetries of some object. As such, we should be able to compose and invert with them. (You can't have both inversion and composition unless you also have a 'do nothing' transformation - the identity.)

There are two standard definitions of group actions. One is as a homomorphism $G\to{\rm Aut}(X)$, where in the category of sets ${\rm Aut}(X)$ just means the permutations of $X$. This encodes the fact that each element of $G$ acts as a permutation of $X$, and that the operation in $G$ corresponds to composition of functions in ${\rm Aut}(X)$. One might suspect we could just say $G\subseteq{\rm Aut}(X)$ is a subgroup of permutations, but this doesn't capture the idea fully: group actions needn't be faithful and so different group elements can behave as the same function on $X$ (i.e. $G\to{\rm Aut}(X)$ is not an injective homomorphism).

The other definition of a group action is as a map $G\times X\to X$ satisfying a particular set of properties. The map $G\times X\to X$ should be thought of as the "evaluation" map wherein $(g,x)$ is sent to $g(x)$ (here we treat $g\in G$ as a function of $X$). And so we want our functions to satisfy "associativity" viz. $f(g(x))=(f\circ g)(x)$; this is what the condition means.