The inversion group consists of only two operations: the identity and the inversion. Inversion is the matrix operation (in 3 spatial dimensions):
which we might also write as . The combination of SO(3) and the inversion symmetry forms O(3), the Orthogonal Group in Three Dimensions, which is the set of all coordinate transformations that leave the length of a vector unchanged.
This is nowhere near all of the groups of interest and use in physics! It isn't even all of the Lie groups of coordinate transformations of interest and use in physics. As we will see in some detail in later chapters, the theory of special relativity is most beautifully defined by looking for the set of transformations of four dimensional spacetime that leave a particular definition of the length of a four-vector invariant, although that is beyond the scope of this introduction or review. Rather, it is the motivation for this review - you won't understand what I'm talking about when I get to the Lorentz group if you don't know what a group is!