nLab central product of groups

Definition

Definition

(central product)

Given two groups $G_1$ and $G_2$ and a joint subgroup

(1)$$C \xhookrightarrow{\iota_i} Z(G_i)$$

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

$$G_1 \circ G_2 \;\coloneqq\; \big( G_1 \times G_2 \big)/_{diag} C$$

of the direct product group $G_1 \times G_2$ by the diagonal subgroup $C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2$.

(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 $C_i \xhookrightarrow{\iota_i} Z(G_i)$

2. 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 $(-)^{-1}\circ \phi$.

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

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).

Related concepts

• direct product of groups

• semidirect product group

• free product of groups

• central extension of groups

References

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

See also

• Wikipedia, Central product

• GroupProps, External central product