Central product

In mathematics, especially in the field of group theory, the central product is one 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.
There are several related but distinct notions of central product. Similarly to the direct product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled.
A group "G" is an internal central product of two subgroups "H", "K" if
Sometimes the stricter requirement that "H" ∩ "K" is exactly equal to the center is imposed, as in . The subgroups "H" and "K" are then called central factors of "G".
The external central product is constructed from two groups "H" and "K",
two subgroups "H"1 ≤ Z("H"), "K"1 ≤ Z("K"), and a group isomorphism "θ":"H"1 → "K"1. The external central product is the quotient of the direct product "H" × "K" by the normal subgroup "N" = { ( "h", "k" ) : "h" in "H"1, "k" in "K"1, and "θ"("h")⋅"k" = 1 }, . Sometimes the stricter requirement that "H"1 = Z("H") and "K"1 = Z("K") is imposed, as in .
An internal central product is isomorphic to an external central product with "H"1 = "K"1 = "H" ∩ "K" and "θ" the identity. An external central product is an internal central product of the images of "H" × 1 and 1 × "K" in the quotient group formula_1. This is shown for each definition in and .
Note that the external central product is not in general determined by its factors "H" and "K" alone. The isomorphism type of the central product will depend on the isomorphism "θ". It is however well defined in some notable situations, for example when "H" and "K" are both finite extra special groups and formula_2 and formula_3.
Applications.
The representation theory of central products is very similar to the representation theory of direct products, and so is well understood, .
Central products occur in many structural lemmas, such as which is used in George Glauberman's result that finite groups admitting a Klein four group of fixed-point-free automorphisms are solvable.