A dinilpotent group is a group that can be written as the product
of two nilpotent
subgroups. There is an extensive literature dealing with such groups (see,
for example,
the recent book of Amberg, Franciosi and de Giovanni [1]).
In 1955, Itô proved in [6]
that the product of two abelian groups is always metabelian, and in the
following
year, Hall and Higman proved, as a special case of [3, Theorem
1.2.4],
that if G=AB is a finite soluble group with A
and
B nilpotent of coprime orders and classes c
and d respectively, then G has derived length at most
c+d. Wielandt [9] showed that
if the finite group G=AB has A and B
nilpotent and of coprime orders, then G is
necessarily soluble. Kegel [7] then showed that
the condition that the orders be
coprime was unnecessary. A natural next question to ask is if the derived
length of
a finite dinilpotent group is bounded by some function of the classes of
the factors,
and, in the light of the Hall–Higman result and the result of Itô,
the following
conjecture seems natural. (The first explicit reference to this conjecture
that we know
of is in Kegel [8]; see also, for example, Problem 5
on
page 36 of [1].)