Contents

group theory

# Contents

## Definition

Given two groups $G_1$ and $G_2$ and a joint subgroup $C \xhookrightarrow{\iota_i} Z(G_i)$ 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 $C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2$.

Beware that there is no widely accepted conventnotation for the central product, and that most notational conventions supporess 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 no symbol at all.

Also beware that most texts insists on stating the choices as that of two separate subgroups $C_i \xhookrightarrow{\iota_i} G_i$ together with an isomorphism $C_1 \xrightarrow[\simeq]{\phi} C_2$ between them.

## 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)