Singer-Zyklen in klassischen Gruppen
About: This article is published in Mathematische Zeitschrift.The article was published on 1970-03-01. It has received 90 citations till now.
Citations
More filters
Book•
01 Jul 2013TL;DR: In this paper, the maximal subgroups of almost simple finite classical groups in dimension up to 12 were classified and the maximal subsets of the almost simple groups with socle one of Sz(q), G2 (q), 2G2(q) or 3D4(q).
Abstract: This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.
395 citations
TL;DR: In this article, the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T was shown to be at most 4.
Abstract: We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many excep- tions, the maximum element order is at most m(T) Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4 These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4 We note again that this result gives upper bounds for meo(Aut(T)) in terms of m(T), and for meo(G) in terms of m(G) (since m(T) ≤ m(G)) Moreover equality in the up- per bound meo(Aut(T)) ≤ m(T) holds when T = PSLd(q) for all but two pairs (d,q), see Table 3 and Theorem 216 (Theorem 216 and Table 3 provide good estimates for
66 citations
TL;DR: In this article, it was shown that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and precisely the parameters for which the graphs are connected, or equivalently, the schemes are primitive.
Abstract: The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.
58 citations
Posted Content•
TL;DR: In this paper, it was shown that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and precisely the parameters for which the graphs are connected, or equivalently, the schemes are primitive.
Abstract: The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.
56 citations
Additional excerpts
...Also A1 contains the irreducible element ω̂ of order p −1 p−1 , whereas (see [1] or [10]) for the remaining groups Y , |〈ω̂〉 ∩ NGL(a,q0)(Y )| ≤ (q a/2 0 + 1)(q0 − 1)....
[...]
TL;DR: In this paper, the authors present polynomial-time algorithms for finding generators for a Sylow p-subgroup of a set of permutations of an n-set.
55 citations
References
More filters