{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T16:18:26Z","timestamp":1769876306442,"version":"3.49.0"},"reference-count":27,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":396,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2009]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A<jats:italic>p<\/jats:italic>-regular element in a finite group is an element of order not divisible by the prime number<jats:italic>p<\/jats:italic>. We show that for every prime<jats:italic>p<\/jats:italic>and every finite simple group<jats:italic>S<\/jats:italic>, a fair proportion of elements of<jats:italic>S<\/jats:italic>is<jats:italic>p<\/jats:italic>-regular. In particular, we show that the proportion of<jats:italic>p<\/jats:italic>-regular elements in a finite classical simple group (not necessarily of characteristic<jats:italic>p<\/jats:italic>) is greater than 1\/(2<jats:italic>n<\/jats:italic>), where<jats:italic>n<\/jats:italic>\u2013 1 is the dimension of the projective space on which<jats:italic>S<\/jats:italic>acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1\/15. For the alternating group A<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, this proportion is at least 26\/(27\u221a<jats:italic>n<\/jats:italic>), and for sporadic simple groups, at least 2\/29.<\/jats:p><jats:p>We also show that for an arbitrary field<jats:italic>F<\/jats:italic>, if the simple group<jats:italic>S<\/jats:italic>is a quotient of a finite subgroup of<jats:italic>GL<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>F<\/jats:italic>) then for any prime<jats:italic>p<\/jats:italic>, the proportion of<jats:italic>p<\/jats:italic>-regular elements in<jats:italic>S<\/jats:italic>is at least min{1\/31, 1\/(2<jats:italic>n<\/jats:italic>)}.<\/jats:p><jats:p>Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups,<jats:italic>p<\/jats:italic>-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.<\/jats:p><jats:p>Finally we complement our lower bound results with the following upper bound: for all<jats:italic>n<\/jats:italic>\u2265 2 there exist infinitely many prime powers<jats:italic>q<\/jats:italic>such that the proportion of elements of odd order in<jats:italic>PSL<\/jats:italic>(<jats:italic>n,q<\/jats:italic>) is less than 3\/\u221a<jats:italic>n<\/jats:italic>.<\/jats:p>","DOI":"10.1112\/s1461157000000036","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T11:42:44Z","timestamp":1375789364000},"page":"82-119","source":"Crossref","is-referenced-by-count":31,"title":["On the Number of<i>p<\/i>-Regular Elements in Finite Simple Groups"],"prefix":"10.1112","volume":"12","author":[{"given":"L\u00e1szl\u00f3","family":"Babai","sequence":"first","affiliation":[]},{"given":"P\u00e9ter P.","family":"P\u00e1lfy","sequence":"additional","affiliation":[]},{"given":"Jan","family":"Saxl","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000000036_ref007","volume-title":"ATLAS of Finite Groups","author":"Conway","year":"1985"},{"key":"S1461157000000036_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107360228.004"},{"key":"S1461157000000036_ref026","volume-title":"The geometry of the classical groups","author":"Taylor","year":"1992"},{"key":"S1461157000000036_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/BF01263536"},{"key":"S1461157000000036_ref016","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511629235"},{"key":"S1461157000000036_ref015","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.2001.9016"},{"key":"S1461157000000036_ref014","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1995.1238"},{"key":"S1461157000000036_ref024","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700001191"},{"key":"S1461157000000036_ref010","article-title":"A Generating Function Approach to the Enumeration of Matrices in Groups over Finite Fields","volume":"176","author":"Fulman","year":"2005","journal-title":"Mem. Amer. Math. Soc."},{"key":"S1461157000000036_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0081548"},{"key":"S1461157000000036_ref004","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1515\/9783110872743.39","volume-title":"Groups and Computation","volume":"8","author":"Babai","year":"2001"},{"key":"S1461157000000036_ref003","unstructured":"3. Babai L. , Beals R. and Seress \u00c1. , \u2018Polynomial-time theory of matrix groups\u2019, Proc. 41st ACM Symp. Theory of Computing (STOC'09) (ACM Press), to appear."},{"key":"S1461157000000036_ref027","doi-asserted-by":"publisher","DOI":"10.1007\/BF01692444"},{"key":"S1461157000000036_ref025","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0081546"},{"key":"S1461157000000036_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01109836"},{"key":"S1461157000000036_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2005.10.011"},{"key":"S1461157000000036_ref017","unstructured":"17. Lawther R. , personal communication, April 2008."},{"key":"S1461157000000036_ref023","doi-asserted-by":"publisher","DOI":"10.1112\/S0024611598000422"},{"key":"S1461157000000036_ref001","doi-asserted-by":"crossref","unstructured":"1. Babai L. and Szemer\u00e9di E. , \u2018On the complexity of matrix group problems I\u2019, Proc. 25th IEEE Symp. on Theory of Computing (FOCS'84) (IEEE Comp. Soc. Press, 1984) 229\u2013240.","DOI":"10.1109\/SFCS.1984.715919"},{"key":"S1461157000000036_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(74)90150-1"},{"key":"S1461157000000036_ref009","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1978-092-5"},{"key":"S1461157000000036_ref019","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-65.2.297"},{"key":"S1461157000000036_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548302005217"},{"key":"S1461157000000036_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF02020968"},{"key":"S1461157000000036_ref022","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.2000.8548"},{"key":"S1461157000000036_ref011","unstructured":"11. Gager P. C. , \u2018Maximal tori in finite groups of Lie type\u2019, PhD Thesis, University of Warwick, 1973."},{"key":"S1461157000000036_ref012","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1515\/9783110872743.169","volume-title":"Groups and Computation III","author":"Guralnick","year":"2001"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000000036","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,3]],"date-time":"2023-07-03T18:17:49Z","timestamp":1688408269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000000036\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009]]},"references-count":27,"alternative-id":["S1461157000000036"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000000036","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009]]}}}