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CMA Research Report


MRR03-016

M.F. Newman, E.A. O'Brien and M.R. Vaughan-Lee

Groups and nilpotent Lie rings whose order is the sixth power of a prime


Abstract:  We have proved that there are

3p2 + 39p + 344 + 24 gcd(p - 1,3) + 11 gcd(p - 1,4) + 2 gcd(p - 1,5)

isomorphism types of groups and nilpotent Lie rings with order p6 for every prime p ≥ 5. We establish the result, and power-commutator presentations for the groups, in various ways. The most novel method constructs product presentations for nilpotent Lie rings with order p6 and then uses the Baker-Campbell-Hausdorff formula to construct the power-commutator presentations for the corresponding groups. Public access to the group presentations is provided via a database distributed with computer algebra systems.


AMS Classification:  20D15
Date:  8 December 2003

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