Open Access
2007 On the Efficiency of the Simple Groups of Order Less Than a Million and Their Covers
Colin M. Campbell, George Havas, Colin Ramsay, Edmund F. Robertson
Experiment. Math. 16(3): 347-358 (2007).


There is much interest in finding short presentations for the finite simple groups. In a previous paper we produced nice efficient presentations for all small simple groups and for their covering groups. Here we extend those results from simple groups of order less than100,000 up to order one million, but we leave one simple group and one covering group for which the efficiency question remains unresolved. We give presentations that are better than what was previously available, in terms of length and in terms of computational properties, in the process answering two previously unresolved problems about the efficiency of covering groups of simple groups. Our results are based on major amounts of computation. We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration.


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Colin M. Campbell. George Havas. Colin Ramsay. Edmund F. Robertson. "On the Efficiency of the Simple Groups of Order Less Than a Million and Their Covers." Experiment. Math. 16 (3) 347 - 358, 2007.


Published: 2007
First available in Project Euclid: 7 March 2008

MathSciNet: MR2367323
zbMATH: 1133.20015

Primary: 20-04 , 20F05
Secondary: 20D05 , 20D06 , 20D08

Keywords: covering groups , Efficient presentations , simple groups

Rights: Copyright © 2007 A K Peters, Ltd.

Vol.16 • No. 3 • 2007
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