| Abstract:
It was shown by H. Schröter [Nachr. Ges. Wiss. Göttingen 1889,
193--236] that one combinatorially possible (10,3) design cannot be
realized in a projective plane over any field. In 1954, R. Lauffer
[Math. Nachrichten, vol. 11] gave a representation of this design
over the infinite division ring of quaternions. This proves that the
design, viewed as a ternary implicational system, is strictly
consistent, meaning that it is impossible to derive x=y from the ten
defining relations. Here we give an elementary algebraic proof of
Schröter's theorem. We also give a group representation of rank 3 for
this configuration and show that 3 is the minimal possible such
representation. This is part of an unpublished set of notes on group
representations written in collaboration with Late Professor
Mendelsohn and Barry Wolk.
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