@Leaky Sure, I'm not talking on the level of being technically imprecise, but mentally we can all fill in the quantifiers and infer the only thing it'd be reasonable for him to ask
Conjecture: There exists a bijection between the states of a geometric object and the set of actions on that object. where the actions have some restrictions they must turn one state into another So, I cannot find a non-trivial stabalizer.
Ah you guys spoiled the exercise. But yeah the linear transformations that preserve lines are stretching/squeezing, meaning scalar multiples of the identity
So while @Jacksoja gave me a permutation that was in the stabalizer, that was not a valid action that turned one state of the square into another like switching 2 and 3
"the way i'd put it is that we can indicate a configuration of the circle by marking a point on it and then the set of possible configurations of the circle is just parametrized by the circle itself"