I was studying from Vinberg’s “A Course In Algebra” and while discussing operations on algebraic structures, he says :
For example, according to the axioms of Euclidean geometry, the product of two motions of the plane is again a motion.
Can anyone clarify what he means by product of motions of a plane, and how the end result is again a motion?
A motion of the plane will be a translation, rotation, reflection, or glide reflection.
The product of two motions will involve doing one motion and then do the other motion on the result. The combination will itself be a motion.