TABLE OF CONTENTS We show in this chapter that gyrogroups that are equipped with the so called gyrofactor admit special extensions, giving rise to new gyrogroups that admit the notion of inner product and norm, and possess transformation groups that keep their inner product and norm invariant. In gyrolanguage, these inner product and norm are called gyroinner product and gyronorm.
To appreciate the usefulness of the gyrogroup extension that we study in this chapter we may note that the gyrogroup extension of the Einstein relativistic gyrogroup ( 
, ? E) of all relativistically admissible velocities gives rise to the gyrocommutative gyrogroup of "Lorentz boosts". A Lorentz boost, in turn, is a "Lorentz transformation without rotation" in the jargon. The Lorentz transformation group of spacetime in special relativity theory will turn out in Chap. 10 to be a gyrosemidirect product group of the gyrogroup of Lorentz boosts and a group of space rotations.
Let ? be a positive function, ? : G ? ? >0, of a gyrogroup ( G, ?). The function ?( ?) , ? ? G, is called a gyrofactor of the gyrogroup ( G, ?) if it satisfies the following conditions:
? is normalized, ?(0) = 1, where 0 is the identity element of G.
? is even, ?( ? ?) = ?( ?) for all ? ? G.
? is gyroinvariant, that is,
