  The new local group LB1 introduced previously will be studied in detail,
depicting its unique nature that makes it a new group in fundamental physics.
It will be made clear that even though most of its elements are Lorentz
transformations, one unique discrete transformation not present in the Lorentz
groups, is making this group into a new group. In fact there will be two kinds
of discrete transformations, one a Lorentz transformation, the other not. It is
these discrete transformations that allow for an isomorphism between the group
SO(2) and LB1. An isomorphism between the compact group SO(2) and the
non-compact group SO(1,1) plus two kinds of discrete transformations that make
up the new group LB1.
