bi-gyrogroup: the group-like structure induced by bi-decomposition of groups
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abstract
the decomposition $gamma=bh$ of a group $gamma$ into a subset $b$ and a subgroup $h$ of $gamma$ induces, under general conditions, a group-like 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 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 pseudo-euclidean spaces of signature $(1, n)$. the study in this article is motivated by generalized lorentz groups $so{m, n}$, $m, ninn$, in pseudo-euclidean spaces of signature $(m, n)$. accordingly, this article explores the bi-decomposition $gamma = h_lbh_r$ of a group $gamma$ into a subset $b$ and subgroups $h_l$ and $h_r$ of $gamma$, along with the novel bi-gyrogroup structure of $b$ induced by the bi-decomposition of $gamma$. as an example, we show by methods of clifford algebras that the quotient group of the spin group $spin{m, n}$ possesses the bi-decomposition structure.
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mathematics interdisciplinary researchجلد ۱، شماره ۱، صفحات ۱۱۱-۱۴۲
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