By Hatcher A.

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32. −(a for any a, b∈G. 132) 30 CHAPTER 1. GYROGROUPS Proof. 133) [a, −b]{−(−gyr[a, −b]b − a)} (6) === −(−b − gyr−1 [a, −b]a) (7) === −{−b − gyr[b, −a]a} (8) === −{(−b) (−a)} . 132). 133) follows. (1) Follows from Def. 9, p. 7, of the gyrogroup cooperation . 105). 13(12) applied to the term {. } in (2). 13(12) applied to b, that is, gyr[a, −b]b = −gyr[a, −b](−b). 27. (6) Follows from (5) by distributing the gyroautomorphism gyr−1 [a, −b] over each of the two terms in {. }. 106). (8) Follows from (7) by Def.

71) for all a, b ∈ G. 72) 20 CHAPTER 1. GYROGROUPS for all a, b ∈ G. 73) = (b a)⊕gyr[b a, a]a = (b a) a, where we employ the left gyroassociative law, the left loop property, and the deﬁnition of the gyrogroup cooperation. 72) form the three basic cancellation laws of gyrogroup theory. Indeed, these cancellation laws are used frequently in the study of gyrogroups and gyrovector spaces. 7 COMMUTING AUTOMORPHISMS WITH GYROAUTOMORPHISMS The commutativity between automorphisms and gyroautomorphisms of a gyrogroup is not the ordinary one but, rather, it is a special commutative law.

6 THE BASIC CANCELLATION LAWS OF GYROGROUPS The basic cancellation laws of gyrogroup theory are obtained in this section from the basic equations of gyrogroups solved in Sec. 5. 13(9). 71) for all a, b ∈ G. 72) 20 CHAPTER 1. GYROGROUPS for all a, b ∈ G. 73) = (b a)⊕gyr[b a, a]a = (b a) a, where we employ the left gyroassociative law, the left loop property, and the deﬁnition of the gyrogroup cooperation. 72) form the three basic cancellation laws of gyrogroup theory. Indeed, these cancellation laws are used frequently in the study of gyrogroups and gyrovector spaces.