nLab central product of groups

Definition

Definition

Given a pair of groups, $G_1$ and $G_2$, and a joint subgroup

(1)$$C \xhookrightarrow{\iota_i} Z(G_i) \,,\;\; i \in \{ 1,2 \}$$

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 $C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2$.

(Dummit & Foote 2003 p. 157)

Remark

(structural over material definition)

Beware that texts such as Gorenstein 1980 p. 29 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

In Riemannian geometry and spin geometry:

Example

A Spin^c-group is a central product of a spin group with the circle group.

Example

The groups Sp(n).Sp(1) and Spin(n).Spin(m) are central products of quaternionic unitary groups and of spin groups, respectively.

Related concepts

• direct product of groups

• semidirect product group

• free product of groups

• central extension of groups

References

• Daniel Gorenstein; p. 29, Thm. 5.3 of: Finite Groups, New York (1980)

• M. Aschbacher; (11.1) in: Finite Group Theory, Cambridge University Press (2012) [ISBN:9781139175319, doi:10.1017/CBO9781139175319]

• David S. Dummit, Richard M. Foote; Ex. 12 on p. 157 of: Abstract Algebra, Wiley (2003) [ISBN:978-0-471-43334-7, pdf]

See also:

• Wikipedia, Central product

• GroupProps, External central product

For more references see at Sp(n).Sp(1).