Random generation and chief length of finite groups

Finite groups;Combinatorial probabilities;Group theory--Generators

Menezes, Nina E. ; Quick, M. R. (Martyn R.) ; Quick, M. R. (Martyn R.)

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Finite groups;    Combinatorial probabilities;    Group theory--Generators;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/3578/NinaMenezesPhDThesis.pdf?sequence=6&isAllowed=y

来源: DR-NTU

【 摘 要 】

Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabeliansimple group G with d randomly chosen elements, and extends thisidea to consider the conditional probability P[subscript(G,Soc(G))](d), the probabilityof generating an almost simple group G by d randomly chosen elements,given that they project onto a generating set of G/Soc(G). In particularwe show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90,with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10except for 30 almost simple groups G, and we specify this list and provideexact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showingthat for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150.In Part II we consider a related notion. Given a probability ε, we wishto determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a resultof Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G),the minimal number of generators needed to generate G. We obtain boundson the chief length of permutation groups in terms of the degree n, andbounds on the chief length of completely reducible matrix groups in termsof the dimension and field size. Combining these with existing bounds ond(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completelyreducible matrix groups.