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central product of groups

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Definition

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

(Gornestein 80, p. 29)

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 iι iG iC_i \xhookrightarrow{\iota_i} G_i together with an isomorphism C 1ϕC 2C_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)

See also

Last revised on April 13, 2019 at 09:47:15. See the history of this page for a list of all contributions to it.