Introduction of group action as morphism of groups

Question

The usual definition of a group action is as follows.

Let $G$ be a group and $A$ be a set. An action of $G$ on $A$ is defined to be a map $\rho:G\times A\rightarrow A$ satisfying certain conditions.

I was thinking for someone who is going to do more advanced mathematics, it is useful to know that “(almost) every structure on an object (of a category) would be just a morphism (in that category)”

In this case group action $G\times A\rightarrow A$ should be seen as a group homomorphism $G\rightarrow {\rm ISO}(A)$, where ${\rm ISO}(A)$ is the group whose underlying set is the set of all bijective functions from $A$ to $A$ where the group structure is given by composition.

Any thoughts on why it is not introduced (in general) in such terms?

Answer 1

I prefer just to keep the language as simple as possible and the level of abstraction as low as the circumstances allow when teaching courses for students who see the material for the first time. So in the measure theory course, say, I keep working on metric spaces when I need some topology though what I really need is some separability axiom. IMHO, working with more familiar objects in such situation facilitates understanding and reduces distractions from the main topic (after all, measure theory is not about topological subtleties) and does not reduce generality or obscure "the view from the top" too much (in a sense, it does not reduce it at all: if you clearly spell out every time what properties of the continuous functions on a metric space you use for each construction, then the arguments will be easy to repeat for general topological space once those properties are derived in a non-metric setting).

As to your particular example, it took me a few minutes to fully decipher and comprehend the construct "a group homomorphism G→ISO(A), where ISO(A) is the group whose underlying set is the set of all bijective functions from A to A where the group structure is given by composition". Yes, you can say things this way, but, unless you are going to discuss at least one similar but different situations when "group" is not really a group and "composition" is something else, but all interesting theorems and constructions can be transferred to the new setting with only minor modifications and look exactly the same from this higher point of view, it would just hang in the air and look to the students about as contrived as "the image of $3$ under the action of a unique linear map $L$ from $\mathbb R$ to itself such that $L(1)=2$" instead of the usual $2\times 3$ despite the fact that once you go from one real variable calculus to the multivariate one, that is exactly the shift in the language and notation from $f'(x)h$ to $(D_xf)h$. Everything should be done at its proper time and for a clear (to the students) reason. So in teaching I would recommend sticking to the good good KISS principle unless you are absolutely forced to deviate from it by the circumstances.

Answer 2

Here are two reasons in favour of the definition as a map $G \times A \to A$ (whether they outweigh the reasons in favour of the group homomorphism definition is of course a judgement call).

• With this definition it is more convenient to introduce topological concepts. Specifically, if $G$ is a topological group and $A$ is a topological space, then one can easily speak of separate and/or joint continuity of the group action $G \times A \to A$. If one, instead, defines a group action as a group homomorphism $G \to \operatorname{ISO}(A)$, this become less intuitive (at least from my perspective).

• One pedagogical advantage of defining group actions as mappings $G \times A \to A$ that satisfy a list of axioms, is that it gives the students additional practice in deriving properties of abstract objects from a list of axioms. (Whether this is really relevant in a particular course depends, of course, on the preliminary knowledge and experience of the students.)