Nonsplit nilpotent group extensions that split at every prime

Abstract

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.