Central product

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

1. G is generated by H and K.
2. Every element of H commutes with every element of K. (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 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 ${\displaystyle H_1\le Z(H)}$ 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 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 ${\displaystyle (H\times K)/N}$. 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 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 ${\displaystyle H_1=Z(H)}$ and ${\displaystyle K_1=Z(K)}$.