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Heisenberg groups, semifields, and translation planes. (English) Zbl 1359.12005

Summary: For Heisenberg groups constructed over semifields (i.e., not necessarily associative division rings), we solve the isomorphism problem and determine the automorphism groups. We show that two Heisenberg groups over semifields are isomorphic precisely if the semifields are isotopic or anti-isotopic to each other. This condition means that the corresponding translation planes are isomorphic or dual to each other.

MSC:

12K10 Semifields
17A35 Nonassociative division algebras
20D15 Finite nilpotent groups, \(p\)-groups
20F28 Automorphism groups of groups
51A35 Non-Desarguesian affine and projective planes
51A10 Homomorphism, automorphism and dualities in linear incidence geometry
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