Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts Given a pair of groups, and , and a joint subgroup in each of their centers, then the corresponding (“external”) central product is the quotient group of the direct product group by the diagonal subgroup . (structural over material definition) Beware that most texts insists on stating the choices in Def. as that of two separate subgroups an isomorphism between them and insists that the second groups as via These clauses matter if one thinks of the subgroup inclusions as in material set theory. But we speak structural set theory, which means that a subgroup inclusion as in (1) is really a choice of monic homomorphism, and this choice already absorbs the choice of and or of . (notation) Beware that there is no widely accepted convention for the notation of central products, and that most notational conventions suppress the choices of central subgroups involved. The “”-notation is popular in finite group-theory, while in Riemannian geometry people tend to use “” (see Sp(n).Sp(1)) or just plain juxtaposition, with no symbol for the central product at all. A Spin^c-group is a central product of a spin group with the circle group. Moreover, in Riemannian geometry and spin geometry one considers the central products Sp(n).Sp(1) and Spin(n).Spin(m). direct product of groups semidirect product group free product of groups central extension of groups See also: Wikipedia, Central product GroupProps, External central product For more references see at Sp(n).Sp(1).