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. Sometimes the stricter requirement that H\cap 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 H_1 \le Z(H) and K_1 \le 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 }, . Sometimes the stricter requirement that H1 = Z(H) and K1 = Z(K) 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 (H\times K) / N. 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 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, .

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.

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 .

References