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Nonsplit nilpotent group extensions that split at every prime



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.


Nonsplit nilpotent group extensions that split at every prime



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