Internal central product

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Suppose G is a group and H and K are subgroups of G. We say that G is an internal central product of H and K if both the following conditions are satisfied:

  1. Every element of H commutes with every element of K, i.e., the subgroups centralize each other.
  2. G = HK, i.e., G is the product of the two subgroups.

Note that in this case, both H and K are central factors of G.

There is a corresponding external notion: external central product.