Prove that all rotations and translations form a subgroup of the group of all reflections and products of reflections in Euclidean Geometry. What theorems do we use to show that this is a subgroup?

I know that I need to show that the subset is

closed

identity is in the subset

every element in the subset has an inverse in the subset.

I don't have to prove associative property since that is already proven with Isometries. What theorems for rotations and translations so that they are closed, identity is in the subset and every element is the subset has an inverse in the subset.