All groups and rings in this paper are finite, and p denotes a prime number.

R. Shepherd [28] and C. R. Leedham-Green and S. McKay [16] showed that any p-group of maximal class contains a subgroup of class at most 2 the index of which is bounded above by a function of p. These papers gave rise to a program for the classification of p-groups that used the notion of coclass proposed by C. R. Leedham-Green and M. F. Newman [21] in 1980. They made several conjectures. The strongest conjecture, Conjecture A, asserted that every p-group of coclass r contains a subgroup of class at most 2 the index of which is bounded above by a function depending only on p and r. These conjectures were proved in a long series of papers by C. R. Leedham-Green, S. McKay, S. Donkin, W. Plesken, A. Shalev and E. Zelmanov (cf. [3, 16–20, 27]). In a recent paper, A. Shalev [26] gave a proof of Conjecture A for all primes p with explicit bounds.