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 Sometimes the stricter requirement that
is exactly equal to the center is imposed, as in (Leedham-Green & McKay 2002, p. 32).
Sometimes the stricter requirement that H1 = Z(H) and K1 = Z(K) is imposed, as in (Leedham-Green & McKay 2002, p. 32).
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.
It is however well defined in some notable situations, for example when H and K are both finite extra special groups and
The representation theory of central products is very similar to the representation theory of direct products, and so is well understood, (Gorenstein 1980, Ch.
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,