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.