Central product

In mathematics, especially in the field of group theory, the central product is way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.

Definition

A group "G" is the central product of two of its subgroups "G₁" and "G₂" if the following hold:

* "G = G₁ G₂"
* For all $g\_1\; in\; G\_1$ and $g\_2\; in\; G\_2,\; g\_1\; g\_2\; =\; g\_2\; g\_1$
* $G\_1\; cap\; G\_2$ is contained in the center of "G".

A sufficient condition for "G" to be the central product of "G₁" and "G₂" is that "G" is isomorphic to the direct product of "G₁" and "G₂".

Examples

* Every extraspecial group is a central product of extraspecial groups of order "p"³.

References

*Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 | id=MathSciNet | id = 81b:20002 | year=1980