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 G is generated by H and K and every element ofH commutes with every element of K . 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 H1 ≤ Z, K1 ≤ Z, and a group isomorphismθ:H1 → K1. The external central product is the quotient of the direct product H × K by the normal subgroupN =,. Sometimes the stricter requirement that H1 = Z and K1 = Z is imposed, as in. An internal central product is isomorphic to an external central product with H1 = K1 = 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. 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 extraspecial groups and and.
Every extra special group is a central product of extra special groups of order p3.
The layer of a finite group, that is, the subgroup generated by all subnormalquasisimple subgroups, is a central product of quasisimple groups in the sense of Gorenstein.
Applications
The representationtheory 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.
