nLab central product of groups





(central product)

Given two groups G 1G_1 and G 2G_2 and a joint subgroup

(1)Cι iZ(G i) C \xhookrightarrow{\iota_i} Z(G_i)

in each of their centers, then the corresponding (“external”) central product is the quotient group

G 1G 2(G 1×G 2)/ diagC G_1 \circ G_2 \;\coloneqq\; \big( G_1 \times G_2 \big)/_{diag} C

of the direct product group G 1×G 2G_1 \times G_2 by the diagonal subgroup C(ι 1,ι 2)G 1×G 2C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2.

(Gorenstein 80, p. 29)


(structural over material definition)

Beware that most texts insists on stating the choices in Def. as that of

  1. two separate subgroups C iι iZ(G i)C_i \xhookrightarrow{\iota_i} Z(G_i)

  2. an isomorphism C 1ϕC 2C_1 \xrightarrow[\simeq]{\phi} C_2 between them

and insists that the second groups as via () 1ϕ(-)^{-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 ϕ\phi and or of () 1ϕ(-)^{-1}\circ \phi.



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

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).


  • D. Gorenstein, p. 29 of Finite Groups, New York (1980)

See also

Last revised on May 16, 2023 at 14:09:13. See the history of this page for a list of all contributions to it.