Given two groups $G_1$ and $G_2$ and a joint subgroup
in each of their centers, then the corresponding (“external”) central product is the quotient group
of the direct product group $G_1 \times G_2$ by the diagonal subgroup $C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2$.
(structural over material definition)
Beware that most texts insists on stating the choices in Def. as that of
two separate subgroups $C_i \xhookrightarrow{\iota_i} Z(G_i)$
an isomorphism $C_1 \xrightarrow[\simeq]{\phi} C_2$ between them
and insists that the second groups as via $(-)^{-1}\circ \phi$
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 $\phi$ and or of $(-)^{-1}\circ \phi$.
(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 “$\circ$”-notation is popular in finite group-theory, while in Riemannian geometry people tend to use “$\cdot$” (see Sp(n).Sp(1)) or just plain juxtaposition, with no symbol for the central product at all.
In Riemannian geometry and spin geometry one considers the central products Sp(n).Sp(1) and Spin(n).Spin(m).
See also
Wikipedia, Central product
GroupProps, External central product
Last revised on May 16, 2019 at 09:59:54. See the history of this page for a list of all contributions to it.