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On Mendelsohn's Pathology for Projective and Affine Planes

On Mendelsohn's Pathology for Projective and Affine Planes ON MENDELSOHN'S PATHOLOGY FOR PROJECTIVE AND AFFINE PLANES J. B. WILKER If ( is a line in a projective plane n then by removing «f and the points on it we pass to an affine plane it . The collineation group of n is isoraorphic to the stabilizer e e of / in the collineation group of n. That is, Aut (n/) ~ [Aut (n)]^ In [2] Eric Mendelsohn proved the following remarkable theorem. Given a group G it is possible to construct a projective plane n such that G ~ Aut (ft). It was natural to ask whether the construction of n could be tailored to a given subgroup H c G to achieve G ^ Aut (n) and H ~ Aut (T^) for a suitable ^ = £ (H) in 7r. In [3] Mendelsohn proved that all the normal subgroups of G (including 1 and G) can be represented simultaneously in this way. The purpose of this note is to remove the restriction to normal subgroups. THEOREM. Given a group G there is a projective plane n such that G ^ Aut (n). Moreover, ifH is any subgroup of G there is a line / = £ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Mendelsohn's Pathology for Projective and Affine Planes

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/9.2.163
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Abstract

ON MENDELSOHN'S PATHOLOGY FOR PROJECTIVE AND AFFINE PLANES J. B. WILKER If ( is a line in a projective plane n then by removing «f and the points on it we pass to an affine plane it . The collineation group of n is isoraorphic to the stabilizer e e of / in the collineation group of n. That is, Aut (n/) ~ [Aut (n)]^ In [2] Eric Mendelsohn proved the following remarkable theorem. Given a group G it is possible to construct a projective plane n such that G ~ Aut (ft). It was natural to ask whether the construction of n could be tailored to a given subgroup H c G to achieve G ^ Aut (n) and H ~ Aut (T^) for a suitable ^ = £ (H) in 7r. In [3] Mendelsohn proved that all the normal subgroups of G (including 1 and G) can be represented simultaneously in this way. The purpose of this note is to remove the restriction to normal subgroups. THEOREM. Given a group G there is a projective plane n such that G ^ Aut (n). Moreover, ifH is any subgroup of G there is a line / = £

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1977

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