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The Steiner system S(5,8, 24) constructed from dual affine planes

Published online by Cambridge University Press:  14 November 2011

G. A. Kadir  and
J. D. Key
G. A. Kadir
Affiliation:
Department of Mathematics, University of Mostansryah, College of Education, Baghdad, Iraq
J. D. Key
Affiliation:
Department of Mathematics, University of Birmingham, P.O. Box 363, Birmingham B15 2TT, U.K.
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Synopsis

We construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 1-2 , 1986 , pp. 147 - 160
Copyright
Copyright © Royal Society of Edinburgh 1986

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