Want a quick clarification and a small concrete example of the compatibility property of group action

The Wolfram Mathworld and Wikipedia entries on Group Actions define the compatibility property (ii) of group actions as such:

A group $$G$$ is said to act on a set $$X$$ when there is a map $$G \times X \xrightarrow{\quad \ddagger \quad} X$$ such that condition (ii) holds for all elements $$x \in X$$:

(ii) $$g \ddagger (h \ddagger x) = (g h) \ddagger x$$

for all $$g, h \in G$$.

I a question which I think it trivial and is done merely as a sanity check, and a request for a simple example:

(Question 1): The operation between $$g h$$ is simply whatever operation I had in group $$G$$, yes? Let $$G$$ be the set of complex numbers $$\mathbb{C}$$ augmented with multiplication and have $$g = e^{2i}$$ and $$h = e^{3i}$$. Then we could have $$gh = e^{5i}$$.

(Request:) A simple computation involving some groups $$G$$ and $$X$$ with a concrete representation of what the operation does on the elements. I did not find the answer (https://math.stackexchange.com/a/69677/937672) or related answers in a similar thread satisfactory.

Preferably we'll involve an operation that is around high school or middle school level in terms of computational simplicity and explicitness, using the complex numbers or some subgroup of them if possible.

• I'll try to answer my question in one second with a concrete example, working on it right now.
– Nate
Commented Jan 29 at 13:46

I'm providing this a prospective answer, giving a concrete example of what Adam Saltz wrote in Concrete examples of group actions. To quote Von Neumann, this is the least "baroque" and most "classical" style answer that I can think of:

Let $$X$$ be the set of all 2-dimensional vectors, such that we have some $$x = \left[ \begin{array}{rr} x_1 \\ x_2 \end{array} \right]$$

Now, let $$G$$ be the set of all invertible $$2 \times 2$$ dimensional matrices, such that

$$g = \left[ \begin{array}{rr} g_1 & g_3\\ g_2 & g_4 \end{array} \right]$$

and

$$h = \left[ \begin{array}{rr} h_1 & h_3\\ h_2 & h_4 \end{array} \right]$$

and where $$gh$$ is the usual multiplication of a matrix by another matrix. Let $$\ddagger$$ represent the multiplication of a vector by a matrix.

we can very easily verify for group actions the identity condition (i) such that:

$$e \ddagger x = \left[ \begin{array}{rr} 1 & 0\\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{rr} x_1 \\ x_2 \end{array} \right] = x$$

And with a high school level of knowledge in linear algebra, we know that

$$g \ddagger (h \ddagger x) = \left[ \begin{array}{rr} g_1 & g_3\\ g_2 & g_4 \end{array} \right] \left( \left[ \begin{array}{rr} h_1 & h_3\\ h_2 & h_4 \end{array} \right] \left[ \begin{array}{rr} x_1 \\ x_2 \end{array} \right]\right) = \left( \left[ \begin{array}{rr} g_1 & g_3\\ g_2 & g_4 \end{array} \right] \left[ \begin{array}{rr} h_1 & h_3\\ h_2 & h_4 \end{array} \right] \right) \left[ \begin{array}{rr} x_1 \\ x_2 \end{array} \right] = (gh) \ddagger x$$ which demonstrates property (ii) for compatibility.