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European Mathematical Society Publishing House
2024-03-29 16:51:18
12
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=1&iss=4&update_since=2024-03-29
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2007
4
Editorial
This issue of the journal is published in honor of our teacher, colleague and friend Professor Avinoam Mann. An international conference was held May 14–19, 2006, at the Institute for Advanced Studies of the Hebrew University of Jerusalem in connection with Professor Mann’s retirement from teaching at the Hebrew University after almost 40 years. The conference, devoted to Asymptotic Group Theory, was mainly supported by the Institute forAdvanced Studies, and also by the Einstein Institute of Mathematics of the Hebrew University. It focused on several branches of group theory, such as finite groups and p-groups in particular, profinite groups, residually finite groups, growth and probabilistic aspects – all areas to which Avinoam Mann has made important contributions. The papers in this issue, which were written by friends, colleagues and students of Avinoam Mann, also represent a variety of topics in which his work over the years has made a lasting impact. R. Grigorchuk, A. Lubotzky and A. Shalev
Group theory and generalizations
General
343
346
10.4171/GGD/17
http://www.ems-ph.org/doi/10.4171/GGD/17
Elementary abelian 2-subgroups of Sidki-type in finite groups
Michael
Aschbacher
California Institute of Technology, PASADENA, UNITED STATES
Robert
Guralnick
University of Southern California, LOS ANGELES, UNITED STATES
Yoav
Segev
Ben-Gurion University, BEER-SHEVA, ISRAEL
Finite simple groups, involutions, parabolic subgroups, fundamental subgroups, saturation
Let G be a finite group. We say that a nontrivial elementary abelian 2-subgroup V of G is of Sidki-type in G, if for each involution i in G, CV(i) ≠ 1. A conjecture due to S. Sidki (J. Algebra 39, 1976) asserts that if V is of Sidki-type in G, then V ∩ O2(G) ≠ 1. In this paper we prove a stronger version of Sidki's conjecture. As part of the proof, we also establish weak versions of the saturation results of G. Seitz (Invent. Math. 141, 2000) for involutions in finite groups of Lie type in characteristic 2. Seitz's results apply to elements of order p in groups of Lie type in characteristic p, but only when p is a good prime, and 2 is usually not a good prime.
Group theory and generalizations
General
347
400
10.4171/GGD/18
http://www.ems-ph.org/doi/10.4171/GGD/18
The size of the solvable residual in finite groups
Silvio
Dolfi
Università degli Studi di Firenze, FIRENZE, ITALY
Marcel
Herzog
Tel Aviv University, TEL AVIV, ISRAEL
Gil
Kaplan
The Academic College of Tel Aviv-Yaffo, Tel Aviv, ISRAEL
Arieh
Lev
The Academic College of Tel-Aviv-Yaffo, Tel-Aviv, ISRAEL
Commutator subgroup, centre, Frattini subgroup, Fitting subgroup, solvable residual
Let G be a finite group. The solvable residual of G, denoted by Res(G), is the smallest normal subgroup of G such that the respective quotient is solvable. We prove that every finite non-trivial group G with a trivial Fitting subgroup satisfies the inequality |Res(G)| > |G|β, where β = log(60)/log(120(24)1/3) ≈ 0.700265861. The constant β in this inequality can not be replaced by a larger constant.
Group theory and generalizations
General
401
407
10.4171/GGD/19
http://www.ems-ph.org/doi/10.4171/GGD/19
On kernels of cellular covers
Emmanuel
Farjoun
Hebrew University, JERUSALEM, ISRAEL
Rüdiger
Göbel
Universität Duisburg-Essen, ESSEN, GERMANY
Yoav
Segev
Ben-Gurion University, BEER-SHEVA, ISRAEL
Saharon
Shelah
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Cellular cover, infinite cardinal, free abelian group
In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G → M. We show that in general a torsion free reduced abelian group M may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain “freeness” properties, then its cardinality is bounded by |M|.
Group theory and generalizations
General
409
419
10.4171/GGD/20
http://www.ems-ph.org/doi/10.4171/GGD/20
A partial extension of Lazard's correspondence for finite p-groups
George
Glauberman
University of Chicago, CHICAGO, UNITED STATES
Finite p-groups, Lie algebras
M. Lazard established a correspondence between finite p-groups of nilpotence class less than p and finite nilpotent Lie rings of p-power order and nilpotence class less than p. This correspondence has had many applications, but cannot generally be extended to p-groups of class p or larger. However, in this paper, we obtain a partial extension of Lazard's result for a p-group that is a product of normal subgroups of class less than p.
Group theory and generalizations
General
421
468
10.4171/GGD/21
http://www.ems-ph.org/doi/10.4171/GGD/21
Presentations of finite simple groups: profinite and cohomological approaches
Robert
Guralnick
University of Southern California, LOS ANGELES, UNITED STATES
William
Kantor
University of Oregon, EUGENE, UNITED STATES
Martin
Kassabov
Cornell University, ITHACA, UNITED STATES
Alexander
Lubotzky
Hebrew University, JERUSALEM, ISRAEL
Finite simple groups, generators, relations, presentations, profinite presentations, cohomology, second cohomology group
We prove the following three closely related results: Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. If G is a finite simple group, F a field and M is an FG-module, then dim H2(G,M) ≤ (17.5) dim M. If G is a finite group, F a field and M is an irreducible faithful FG-module, then dim H2(G,M) ≤ (18.5) dim M.
Group theory and generalizations
General
469
523
10.4171/GGD/22
http://www.ems-ph.org/doi/10.4171/GGD/22
Magnus intersections of one-relator free products with small cancellation conditions
Arye
Juhasz
Technion - Israel Institute of Technology, HAIFA, ISRAEL
One-relator free products, small cancellation theory, word-combinatorics, Magnus subgroups
Donald Collins initiated the study of intersections of Magnus subgroups in one-relator groups. In particular, he characterized those intersections of Magnus subgroups that are not Magnus subgroups. In the present work we show that Collins’ results extend to one-relator quotients of free products of groups with a small cancellation condition and give a complete list of those defining relators for which Magnus subgroups do not intersect in a Magnus subgroup. We use van Kampen diagrams and word combinatorics.
Group theory and generalizations
General
525
552
10.4171/GGD/23
http://www.ems-ph.org/doi/10.4171/GGD/23
Permutation groups, minimal degrees and quantum computing
Julia
Kempe
Tel Aviv University, TEL AVIV, ISRAEL
László
Pyber
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Aner
Shalev
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Permutation groups, quantum computing, minimal degree, Hidden Subgroup Problem, characters
We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group H ≤ Sn of minimal degree m and on the number of its elements of any given support. These results contribute to the foundations of a non-commutative coding theory. A main application of our results concerns the Hidden Subgroup Problem for Sn in quantum computing. We completely characterize the hidden subgroups of Sn that can be distinguished from identity with weak Quantum Fourier Sampling, showing that these are exactly the subgroups with bounded minimal degree. This implies that the weak standard method for Sn has no advantage whatsoever over classical exhaustive search.
Group theory and generalizations
Quantum theory
General
553
584
10.4171/GGD/24
http://www.ems-ph.org/doi/10.4171/GGD/24
Complements of the socle in monolithic groups
Andrea
Lucchini
Università di Padova, PADOVA, ITALY
Federico
Menegazzo
Università di Padova, PADOVA, ITALY
Marta
Morigi
Università di Bologna, BOLOGNA, ITALY
Monolithic groups, complements, finite simple groups
We show that if H is a finite group with a unique minimal normal subgroup N, which is not abelian, then the number of conjugacy classes of complements of N in H is strictly smaller than |N|.
Group theory and generalizations
General
585
611
10.4171/GGD/25
http://www.ems-ph.org/doi/10.4171/GGD/25
Growth conditions in infinitely generated groups
Avinoam
Mann
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Polynomial growth
We characterize groups in which each finitely generated subgroup has polynomial growth, under some uniformity conditions.
Group theory and generalizations
General
613
622
10.4171/GGD/26
http://www.ems-ph.org/doi/10.4171/GGD/26
Extremely primitive groups
Avinoam
Mann
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Cheryl
Praeger
The University of Western Australia, CRAWLEY, WA, AUSTRALIA
Ákos
Seress
Ohio State University, COLUMBUS, UNITED STATES
Primitive permutation groups
A primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its orbits. We prove that finite extremely primitive groups are of affine type or almost simple. Moreover, we determine the affine type examples up to finitely many exceptions.
Group theory and generalizations
General
623
660
10.4171/GGD/27
http://www.ems-ph.org/doi/10.4171/GGD/27
Variations on a theme of Burns and Medvedev
Dan
Segal
University of Oxford, OXFORD, UNITED KINGDOM
Relatively free groups, profinite groups
Burns and Medvedev prove in [BM] that a relatively free pro-p group cannot be p-adic analytic unless it is virtually nilpotent. We present a shorter, more conceptual proof, and apply it to deduce analogous results for other categories of groups.
Group theory and generalizations
General
661
668
10.4171/GGD/28
http://www.ems-ph.org/doi/10.4171/GGD/28