Central product

In mathematics, especially in the field of group theory, the central product is 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.

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 and (2) every element of H commutes with every element of K (Gorenstein 1980, p. 29). Sometimes the stricter requirement that H ∩ 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 H₁ ≤ Z(H), K₁ ≤ Z(K), and a group isomorphism θ:H₁ → K₁. The external central product is the quotient of the direct product H × K by the normal subgroup N = { ( h, k ) : where h in H₁, k in K₁, and θ(h)⋅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 × K ) / N. This is shown for each definition in (Gorenstein 1980, p. 29) and (Leedham-Green & McKay 2002, pp. 32–33).

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