Abstract :
The decomposition $Gamma=BH$ of a group $Gamma$ into a subset B and a subgroup $H$ of $Gamma$ induces, under general conditions, a grouplike structure for B, known as a gyrogroup. The famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary mbox{operation} given by the Einstein velocity addition law of special relativity theory. The latter leads to the Lorentz transformation group $so{1,n}$, $ninN$, in pseudoEuclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $so{m, n}$, $m, ninN$, in pseudoEuclidean spaces of signature $(m, n)$. Accordingly, this article explores the bidecomposition $Gamma = H_LBH_R$ of a group $Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of $Gamma$, along with the novel bigyrogroup structure of $B$ induced by the bidecomposition of $Gamma$. As an example, we show by methods of Clifford mbox{algebras} that the quotient group of the spin group $spin{m, n}$ possesses the bidecomposition structure.
