Nice Efficient Presentions for all Small Simple Groups and their Covers

• Published in 2004
• Added on 2026-04-10

Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

Links

• https://www.cambridge.org/core/journals/lms-journal-of-computation-and-mathematics/article/nice-efficient-presentions-for-all-small-simple-groups-and-their-covers/4FB1390E838671B917800302748E127D
• https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4FB1390E838671B917800302748E127D/S1461157000001121a.pdf/div-class-title-nice-efficient-presentions-for-all-small-simple-groups-and-their-covers-div.pdf

Other information

BibTeX entry

@article{NiceEfficientPresentionsforallSmallSimpleGroupsandtheirCovers,
	key = {NiceEfficientPresentionsforallSmallSimpleGroupsandtheirCovers},
	type = {article},
	title = {Nice Efficient Presentions for all Small Simple Groups and their Covers},
	author = {Colin M. Campbell and George Havas and Colin Ramsay and Edmund F. Robertson},
	abstract = {Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.},
	comment = {},
	date_added = {2026-04-10},
	date_published = {2004-04-10},
	urls = {https://www.cambridge.org/core/journals/lms-journal-of-computation-and-mathematics/article/nice-efficient-presentions-for-all-small-simple-groups-and-their-covers/4FB1390E838671B917800302748E127D,https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4FB1390E838671B917800302748E127D/S1461157000001121a.pdf/div-class-title-nice-efficient-presentions-for-all-small-simple-groups-and-their-covers-div.pdf},
	collections = {attention-grabbing-titles,the-groups-group},
	url = {https://www.cambridge.org/core/journals/lms-journal-of-computation-and-mathematics/article/nice-efficient-presentions-for-all-small-simple-groups-and-their-covers/4FB1390E838671B917800302748E127D https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4FB1390E838671B917800302748E127D/S1461157000001121a.pdf/div-class-title-nice-efficient-presentions-for-all-small-simple-groups-and-their-covers-div.pdf},
	year = 2004,
	urldate = {2026-04-10},
	identifier = {doi:10.1112/S1461157000001121},
	journal = {LMS Journal of Computation and Mathematics},
	publisher = {Cambridge University Press},
	volume = 7,
	issn = {1461-1570},
	doi = {10.1112/S1461157000001121},
	pages = {266-283}
}