The purpose of this note is to present examples of nilpotent group
extensions
which split at every p-localization but fail to split. These examples
disprove the
conjecture by Peter Hilton [4, 5] that any nilpotent
group extension with finitely generated quotient which splits at every
prime must necessarily
split. In [3] Hilton
proves his conjecture for extensions with abelian kernel. Moreover, some
further
special cases of the conjecture are established by Casacuberta
and Hilton in [1].
We describe two families of counterexamples to Hilton's conjecture.
In both
families the groups are torsion-free and finitely generated. The first
family consists
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 [3], is the
minimal possible rank for a torsion-free
counterexample to the conjecture. However, this reduction in rank is obtained
at the
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
this crucial
fact together with an outline of its proof.