Constructing normalisers in finite soluble groups

AbstractThis paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser NG(H). If G has composition length n, then H and NG(H) can be constructed using O(n4 log |G|) and O(n5 log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms

Sylow permutable subnormal subgroups of finite groups

Paper published by Elsevier in J. Algebra, 215(2):727-738 (2002). The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1006/jabr.2001.9138[EN] An extension of the well-known Frobenius criterion of p-nilpotence in groups with modular Sylow p-subgroups is proved in the paper. This result is useful to get information about the classes of groups in which every subnormal subgroup is permutable and Sylow permutable.Supported by Proyecto PB97-0674 and Proyecto PB97-0604-C02-02 from DGICYT, Ministerio de Educaci´on y CienciaBallester Bolinches, A.; Esteban Romero, R. (2002). Sylow permutable subnormal subgroups of finite groups. Journal of Algebra. 2(251). doi:10.1006/jabr.2001.9138225

Two matrix group algorithms with applications to computing the automorphism group of a finite p-group

A theoretical description of an algorithm to determine the automorphism group of a finite p-group P was first given by Newman. Implementations of this algorithm with substantial improvements by O’Brien are available in GAP and Magma. The original algorithm, starting with the Frattini quotient V = P/$(P), computes recursively the automorphism group G of the quotient Q of P by successive terms of the lower p-central series of P. Thus the first step returns - G = GL(V). . The heart of the algorithm is the computation of the subgroup of G that normalises a certain subspace of the p-multiplicator M of Q. A refinement in the algorithm replaces G by a subgroup H that normalises certain subspaces of V corresponding to heuristically determined characteristic subgroups of P. In this thesis we describe and give the GAP3 code for two substantial improvements to the algorithm. The first improvement is an algorithm that returns a generating set for the stabiliser in GL(V) of any given sequence of subspaces of a finite dimensional vector space V over any finite field. This is an algorithm of independent interest, as the intersection problem for subgroups of GL(d,pn) is both important and hard. In the algorithm for computing the automorphism group of the p-group P this intersection algorithm is used to compute the precise subgroup K of GL(V) that stabilises the given sequence of subspaces rather than the over-group H of K currently computed. The theoretical basis for the intersection algorithm is a new Galois correspondence between lattices of subspaces of V and subgroups of GL(V). The basic computational tool is the ‘meataxe’ algorithm. As a second contribution, we give an efficient algorithm to compute a canonical form for a subspace U of M under the action of a p-subgroup G of GL(M), and also to compute generators for the subgroup of G that normalises U. Here ‘efficient’ means ‘polynomial in the size of the input’, and M can be any finite dimensional vector space over GF{p). This is important as the kernel of the action of G on V is a p-group; and G itself is often a p-group

Computing automorphism groups and isomorphism testing in finite groups

We outline a new method for computing automorphism groups and performing isomorphism testing for soluble groups. We derive procedures for computing polycyclic presentations for soluble automorphism groups, allowing for much more efficient calculations. Finally, we demonstrate how these methods can be extended to tackle some non-soluble groups. Performance statistics are included for an implementation of these algorithms in the MAGMA [BCP97] language

Verallgemeinerungen der Wielandt-Untergruppe

The Wielandt subgroup and the Norm of a group are group theoretic concepts which have been introduced as generalisations of the center of a group. The first aim of this Thesis is to generalise Wielandt's idea further. For this we first investigate the structural properties of the existing generalisation introduced in my MPhil Thesis. Given a group G this generalisation is a subgroup defined relative to a normal subgroup in G. We investigate the structural behaviour of the generalised Wielandt subgroup and its generalised Wielandt length. We also correct a mistake in my MPhil Thesis. We introduce a new generalisation of the Wielandt subgroup which we call the "relative Wielandt subgroup". It is noted that the relative Wielandt subgroup satisfies some properties which do not hold in case of the ordinary Wielandt subgroup and an example is given which shows that the relative Wielandt subgroup is non-trivial in a wider class of groups than the Wielandt subgroup. Further, we developed algorithms to compute the Norm of a finite or a polycyclic group. We also introduced methods to determine the ordinary, generalised or relative Wielandt subgroup of a finite group. By using these algorithms and the available classification of groups of order dividing p^6, we classify the groups of order dividing p^4 (up to isomorphism) with maximal Wielandt subgroup for all primes and those of order dividing p^6 for a variety of primes.Die Wielandt-Untergruppe und die Norm einer Gruppe sind gruppentheoretische Konzepte, die als Verallgemeinerungen des Zentrums einer Gruppe eingeführt wurden. Das erste Ziel dieser Arbeit ist es die Ideen von Wielandt weiter zu verallgemeinern. Zu diesem Zweck untersuchen wir die strukturellen Eigenschaften der bereits in meiner MPhil-Arbeit eingeführten Verallgemeinerung. Zu einer gegebenen Gruppe G wird diese Verallgemeinerung relativ zu einem Normalteiler in G definiert. Wir untersuchen die strukturellen Eigenschaften dieser verallgemeinerten Wielandt-Untergruppe und die Länge der verallgemeinerten Wielandt-Reihe. Außerdem verbessern wir einen Fehler in der MPhil-Arbeit. Wir führen eine neue Verallgemeinerung der Wielandt-Untergruppe ein, die wir "relative Wielandt-Untergruppe" nennen. Wir bemerken, dass diese relative Wielandt-Untergruppe einige Eigenschaften hat, die für die gewöhnliche Wielandt-Untergruppe nicht gelten. Zusätzlich geben wir ein Beispiel, welches zeigt, dass die relative Wielandt-Untergruppe in einer größeren Klasse von Gruppen nichttrivial ist als w(G). Weiterhin haben wir Algorithmen entwickelt, um die Norm einer endlichen oder polyzyklischen Gruppe G zu berechnen. Wir haben außerdem Methoden entwickelt, um die gewöhnliche, die verallgemeinerte und die relative Wielandt-Untergruppe für eine endliche Gruppe G zu bestimmen. Unter Verwendung dieser Algorithmen und der Klassifikation der Gruppen, deren Ordnung p^6 teilt, klassifizieren wir (bis auf Isomorphie) für alle Primzahlen, die Gruppen mit maximaler Wielandt-Untergruppe, deren Ordnung p^4 teilt. Für die Gruppen, deren Ordnung p^6 teilt, klassifizieren wir diese Gruppen für einige Primzahlen

Hierarchically cocompact classifying spaces for mapping class groups of surfaces

We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group Mod ( S ) of any connected oriented compact surface S , possibly with punctures and boundary components and with negative Euler characteristic has a hierarchically cocompact model for the family of virtually cyclic subgroups of dimension at most vcd Mod ( S ) + 1 . When the surface is closed, we prove that this bound is optimal. In particular, this answers a question of Lück for mapping class groups of surfaces