Involutory decomposition of groups into twisted subgroups and subgroups

An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links bet ween twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are group -like structures which first arose in the study of Einstein's velocity addi tion in the special theory of relativity. In particular, we show that every gyrogroup is a twisted subgroup and that, under general specified conditio ns, twisted subgroups are gyro-commutative gyrogroups. Moreover, we show th at gyrogroups abound in group theory and that they possess rich structure.