Result: SOME REMARKS ON THE PROBABILITY OF GENERATING AN ALMOST SIMPLE GROUP: Some remarks on the probability of generating an almost simple group.

Title:
SOME REMARKS ON THE PROBABILITY OF GENERATING AN ALMOST SIMPLE GROUP: Some remarks on the probability of generating an almost simple group.
Authors:
DALLAVOLTA F., LUCCHINI, ANDREA, MORINI F.
Contributors:
DALLA VOLTA, F, Lucchini, A, Morini, Fiorenza
Source:
Glasgow Mathematical Journal. 45:281-291
Publisher Information:
Cambridge University Press (CUP), 2003.
Publication Year:
2003
Subject Terms:
Generators, relations, and presentations of groups, General Mathematics, finite simple groups, random elements, Probabilistic methods in group theory, Simple groups: alternating groups and groups of Lie type, 0101 mathematics, generators, 01 natural sciences, Arithmetic and combinatorial problems involving abstract finite groups, finite groups, probabylity
Document Type:
Academic journal Article
File Description:
application/xml
Language:
English
ISSN:
1469-509X
0017-0895
DOI:
10.1017/s0017089503001241
Access URL:
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0D16AE35D8B30A947986A726E033B394/S0017089503001241a.pdf/div-class-title-some-remarks-on-the-probability-of-generating-an-almost-simple-group-div.pdf
https://boa.unimib.it/handle/10281/2779
https://www.openaccessrepository.it/record/149968
https://hdl.handle.net/11577/127391
https://doi.org/10.1017/S0017089503001241
Rights:
Cambridge Core User Agreement
Accession Number:
edsair.doi.dedup.....cd81fb9e15460ce6c50b85c20eb4afcd
Database:
OpenAIRE

Further Information

We identify a finite nonabelian simple group \(S\) with the subgroup of inner automorphisms of \(S\) in \(\Aut(S)\). Let \(P(S)\) denote the probability that two randomly chosen elements from \(\Aut(S)\) generate a subgroup containing \(S\), and \(P_S(t)\) be the probability that \(t\) randomly chosen elements from \(S\) generate \(S\). The main result of this paper is that if \(S=\text{PSL}(2,p)\) (where \(p\geq 5\) is a prime), then \(P(S)=P_S(2)+x/|S|\) where \(x\) takes one of four explicit nonnegative values depending on the value of \(p\) modulo \(40\). The proof depends on calculations using \textit{Ph. Hall}'s theorem [Q. J. Math., Oxf. Ser. 7, 134-151 (1936; Zbl 0014.10402)]. Some of these calculations are extended to other finite simple groups using the computer system GAP and give rise to several open questions.