An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are group-like structures which ®rst arose in the study of Einstein's velocity addition in the special theory of relativity. In particular, we show that every gyrogroup is a twisted subgroup and that, under general speci®ed conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure.

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In this note we show that the commutant of a Bol loop of odd order is a subloop.

Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein's velocity addition as a binary operation. Einstein's gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein's addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups and groups implies the existence of a general theory which underlies the theories of groups and gyrogroups and unifies them with respect to their central features. Accordingly, our goal is to construct finite and infinite gyrogroups, both gyrocommutative and non-gyrocommutaive, in order to demonstrate that gyrogroups abound in group theory of which they form an integral part.

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