ALL SIMPLE GROUPS WITH ORDER FROM 1 MILLION TO 5 MILLION ARE EFFICIENT

There is much interest in finding short presentations for the finite simple groups. indeedit has been suggested that all these groups are efficient in a technical sense. in previous papers weproduced nice efficient presentations for all except one of the simple groups with order less than onemillion. here we show that all simple groups with order between 1 million and 5 million are ecient bygiving efficient presentations for all of them. apart from some linear groups these results are all new. wealso show that some covering groups and some larger simple groups are efficient. we make substantialuse of systems for computational group theory and, in particular, of computer implementations of cosetenumeration to find and verify our presentations.