Constructing divisions into power groups

The result, due to Henckell, Margolis, Pin and Rhodes modulo Ash's solution to the pointlike conjecture, that every finite block group divides a power group, has long been considered to be one of the deepest results in finite semigroup and algebraic automata theory. However, the proof is not constructive. Solving a long-standing problem, we provide in this paper an explicit construction of such a division. We also generalize the result to a large class of pseudovarieties of groups. Local group pseudovarieties are also considered, generalizing (and making constructive) results of Margolis and the second author. Some applications to language theory are mentioned.