Random generation and chief length of finite groups
MetadataShow full item record
Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian simple group G with d randomly chosen elements, and extends this idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability of generating an almost simple group G by d randomly chosen elements, given that they project onto a generating set of G/Soc(G). In particular we 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≥10 except for 30 almost simple groups G, and we specify this list and provide exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing that 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 wish to 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 result of 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 bounds on the chief length of permutation groups in terms of the degree n, and bounds on the chief length of completely reducible matrix groups in terms of the dimension and field size. Combining these with existing bounds on d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely reducible matrix groups.
Thesis, PhD Doctor of Philosophy
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Showing items related by title, author, creator and subject.
Plötner, Maria; Over, Harriet; Carpenter, Malinda; Tomasello, Michael (2016-03-24) - Journal articleTo date, developmental research on groups has focused mainly on in-group biases and intergroup relations. However, little is known about children’s general understanding of social groups and their perceptions of different ...
Coutts, Hannah Jane (University of St Andrews, 2011-11) - ThesisPart I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each ...
Cognition in inter-group relations : the effect of group membership on theory of mind and its precursors McClung, Jennifer (University of St Andrews, 2012) - ThesisSocial categorization based on group membership has a significant and broad influence on behaviour (Turner, Hogg, Oakes, Reicher, & Wetherell, 1987). People perceived as being of the same group, ‘in-group’ members, are ...