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HURWITZ GROUPS WITH GIVEN CENTRE

Published online by Cambridge University Press:  24 March 2003

MARSTON CONDER
MARSTON CONDER
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand conder@math.auckland.ac.nz
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Abstract

A Hurwitz group is any non-trivial finite group that can be (2,3,7)-generated; that is, generated by elements $x$ and $y$ satisfying the relations $x^2 = y^3 = (xy)^7 = 1$. In this short paper a complete answer is given to a 1965 question by John Leech, showing that the centre of a Hurwitz group can be any given finite abelian group. The proof is based on a recent theorem of Lucchini, Tamburini and Wilson, which states that the special linear group ${\rm SL}_n(q)$ is a Hurwitz group for every integer $n \geqslant 287$ and every prime-power $q$.

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Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 6 , November 2002 , pp. 725 - 728
Copyright
© The London Mathematical Society 2002

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