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

G is generated by H and K.
Every element of H commutes with every element of K. (Gorenstein 1980, p. 29)

Sometimes the stricter requirement that $H\cap K$ is exactly equal to the center is imposed, as in (Leedham-Green & McKay 2002, p. 32). 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 $H_{1}\leq Z(H)$ and $K_{1}\leq Z(K)$ , and a group isomorphism $\theta \colon H_{1}\to K_{1}$ . The external central product is the quotient of the direct product $H\times K$ by the normal subgroup

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 that H₁ = Z(H) and K₁ = Z(K) is imposed, as in (Leedham-Green & McKay 2002, p. 32).

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 $(H\times K)/N$ . 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 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 $H_{1}=Z(H)$ and $K_{1}=Z(K)$ .

Examples

The Pauli group is the central product of the cyclic group $C_{4}$ and the dihedral group $D_{4}$ .
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 subnormal quasisimple subgroups, is a central product of quasisimple groups in the sense of Gorenstein.

Applications

The representation 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 in George Glauberman's result that finite groups admitting a Klein four group of fixed-point-free automorphisms are solvable.

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, $\S$ 4).

References

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