I am studying " Abstract Algebra " written by dummit/foote. In page 23, This book defines the concept of dihedral group as follows:
For each n = 3, 4, 5, etc... , The set of symmetries of a regular n-gon, where a symmetry is any rigid motion of the n-gon which can be effected by taking a copy back on the original n-gon so it exaclty covers it.
I can understand what it means but I have some curious about this definition because this definition is not rigorous for me. What is the definition of rigid motion? what is copy back? Can we admit this kind of vague terminology in math?
I tried to find other book which describing the concept symmetry. In the book " A first course in Abstract Algebra, this book defines a symmetry of geometrical figure as a rearrangement of the figure preserving the arrangement of its sides and vertices as well as its distance and angles. I thought that this definition is also depending on our intuition... and not rigourous if we compare this with set theory, real analysis, or other mathematical definition...
So, In summary... Q1. I understood that symmetry is a transformation which preserves shape, angle, distances... in intuitive meaning Did I understand well? Q2. If I understood well, what is a precise and rigorous definition of symmetry? as we did in set theory, real analysis, etc... Q3. Can we deal geometry by using a precise and rigorous method only using axiom, set, logic ??