Abstract
An involutory decomposition is a decomposition, due to an involution, of a groupinto a twisted subgroup and a subgroup. We study unexpected links between twisted subgroupsand gyrogroups. Twisted subgroups arise in the study of problems in computational complex-ity. In contrast, gyrogroups are group-like structures which Ærst arose in the study of Einstein'svelocity addition in the special theory of relativity. In particular, we show that every gyrogroupis a twisted subgroup and that, under general speciÆed conditions, twisted subgroups are gyro-commutative gyrogroups. Moreover, we show that gyrogroups abound in group theory andthat they possess rich structure.