nLab
central product of groups

Contents

Contents

Definition

Definition

(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)

Remark

(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.

Remark

(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.

Examples

In Riemannian geometry and spin geometry

In Riemannian geometry and spin geometry one considers the central products Sp(n).Sp(1) and Spin(n).Spin(m).

References

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

See also

Last revised on May 16, 2019 at 09:59:54. See the history of this page for a list of all contributions to it.