What exactly is rigid body motion and symmetry in abstract algebra in terms of function?

Question

My text book of abstract algebra states two definitions of rigid body motion and symmetry as following.

1. Rigid body motion : Rigid body motion of an object $X$ is a bijection from plane to plane s.t distance between $f(x)$ and $f(y)$ (which I think as new assigned value to object $X$) is same as distance between $x$ and $y$ . First of all I have doubt about how we actually represent an object say $X$ on plane what this set $X$ contains?(I think its contains ordered pair but for sake of simplicity we just label these ordered pair by say 1,2 etc.) .Second doubt is there any criteria set $X$ must follow in order to represent an object say a triangle . Lastly why rigid body motion is bijection(the reason I can think of is an object can be any where on the plane so bijection can't be for specific region if it is so what happens to the points other than point in set $X$ or purpose of being bijective function is to just assign new values to object $X$ s.t if object after action/rigid body motion represented by another set say $Y$ then $f(X)=Y$ .
2. Symmetry : Symmetry of an object say $Y$ is bijection from plane to plane s.t $f(Y) = Y$. It basically says new assigned value to object Y must belongs to the set $Y$ . My doubt is why this function(symmetry) is bijection from plane to plane , as $f(Y)=Y$ implies function(symmetry) is bijection from $Y$ to $Y$.(this is the definiton of permutation). So I must have misunderstood something.

Also give some reference on resourses to study this topic. As I may have interpreted everything wrong.

Answer 1

Any subset of the plane can be $X$ so if you want a triangle just pick your points and you've got one. A rigid body motion must both preserve distances and be bijective. The distances are what keeps them rigid and they are bijective because the transformations are invertible, since we can undo them.

Symmetries are a transformation that leaves some subset invariant, but it may move the points around. Say we have an empty square centered at the origin and we consider a quarter turn rotation symmetry. This will leave the square invariant as a set even though the position of the points has changed under the transformation. A similar situation occurs if you consider a reflection symmetry, where most of the points move but it's still a square afterwards. Again, these transformations are invertible, and so bijective. This will be useful because transformations with an identity and inverses will form an group.

My first introduction to symmetries was in "Abstract Algebra" by Dummit and Foote. Studying isometries and rigidity required some knowledge of metric spaces and linear algebra but the group theory I had first learned in this text was the most useful in studying these problems. Examples motivated by the wallpaper groups and crystallography helped me gain an intuition for the subject.