Inmathematics, especially in the field ofgroup theory, thecentral product is one way of producing agroup from two smaller groups. The central product is similar to thedirect product, but in the central product twoisomorphiccentralsubgroups 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 classifyextraspecial groups.

There are several related but distinct notions of central product. Similarly to thedirect product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled.

A groupG is aninternal central product of two subgroupsH,K if

1. G is generated byH andK.
2. Every element ofH commutes with every element ofK. (Gorenstein 1980, p. 29)

Sometimes the stricter requirement that${\displaystyle H\cap K}$ is exactly equal to the center is imposed, as in (Leedham-Green & McKay 2002, p. 32). The subgroupsH andK are then called central factors ofG.

Theexternal central product is constructed from two groupsH andK, two subgroups${\displaystyle H_1\le Z(H)}$ and${\displaystyle K_1\le Z(K)}$, and a group isomorphism${\displaystyle \theta :H_1\to K_1}$. The external central product is the quotient of the direct product${\displaystyle H\times K}$ by the normal subgroup

${\displaystyle N=\{(h,k):h\in H_1,k\in K_1,\text{ and }\theta (h)\cdot k=1\}}$,

(Gorenstein 1980, p. 29). Sometimes the stricter requirement thatH₁ = Z(H) andK₁ = Z(K) is imposed, as in (Leedham-Green & McKay 2002, p. 32).

An internal central product is isomorphic to an external central product withH₁ =K₁ =H ∩K andθ the identity. An external central product is an internal central product of the images ofH × 1 and 1 ×K in the quotient group${\displaystyle (H\times K)/N}$. This is shown for each definition in (Gorenstein 1980, p. 29) and (Leedham-Green & McKay 2002, pp. 32–33).

Note that the external central product is not in general determined by its factorsH andK alone. The isomorphism type of the central product will depend on the isomorphismθ. It is however well defined in some notable situations, for example whenH andK are both finiteextra special groups and${\displaystyle H_1=Z(H)}$ and${\displaystyle K_1=Z(K)}$.

• ThePauli group is the central product of thecyclic group${\displaystyle C_4}$ and thedihedral group${\displaystyle D_4}$.
• Everyextra special group is a central product of extra special groups of orderp³.
• The layer of a finite group, that is, the subgroup generated by allsubnormalquasisimple subgroups, is a central product of quasisimple groups in the sense of Gorenstein.

Therepresentation theory of central products is very similar to the representation theory of direct products, and so is well understood, (Gorenstein 1980, Ch. 3.7).

Central products occur in many structural lemmas, such as (Gorenstein 1980, p. 350, Lemma 10.5.5) which is used inGeorge Glauberman's result that finite groups admitting aKlein four group of fixed-point-free automorphisms aresolvable.

In certain context of a tensor product of Lie modules (and other related structures), the automorphism group contains a central product of the automorphism groups of each factor (Aranda-Orna 2022,${\displaystyle \mathrm{\S }}$ 4).

• Gorenstein, Daniel (1980),Finite Groups, New York: Chelsea,ISBN 978-0-8284-0301-6,MR 0569209
• Leedham-Green, C. R.; McKay, Susan (2002),The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27,Oxford University Press,ISBN 978-0-19-853548-5,MR 1918951
• Aranda-Orna, Diego (2022),On the Faulkner construction for generalized Jordan superpairs, Linear Algebra and its Applications, vol. 646, pp. 1–28

• Finite groups