Black box group

The upper bound on the order of G given by |G| ≤ 2N shows that G is finite.

The black box groups were introduced by Babai and Szemerédi in 1984.

[1] They were used as a formalism for (constructive) group recognition and property testing.

Many other algorithms require finding element orders.

Since there are efficient ways of finding the order of an element in a permutation group or in a matrix group (a method for the latter is described by Celler and Leedham-Green in 1997), a common recourse is to assume that the black box group is equipped with a further oracle for determining element orders.