The purpose of this note is to present examples of nilpotent group
which split at every p-localization but fail to split. These examples
conjecture by Peter Hilton [4, 5] that any nilpotent
group extension with finitely generated quotient which splits at every
prime must necessarily
split. In  Hilton
proves his conjecture for extensions with abelian kernel. Moreover, some
special cases of the conjecture are established by Casacuberta
and Hilton in .
We describe two families of counterexamples to Hilton's conjecture.
families the groups are torsion-free and finitely generated. The first
of groups of class 2 and rank 6. In the second family, the groups have
rank 5, which,
in view of Hilton's result in , is the
minimal possible rank for a torsion-free
counterexample to the conjecture. However, this reduction in rank is obtained
expense of increasing the class from 2 to 3. Theorem 2·3 establishes
that this increase
in class is unavoidable: there are no torsion-free counterexamples of rank
5 and class 2.
The proof of Theorem 2·3 is based on a fact, Lemma 2·4,
about quadratic forms.
The author would like to thank J. W. S. Cassels for providing him with
fact together with an outline of its proof.