Schedule for Glauberman Conference [0]
March 24-28, 2008
University of Chicago

PRINCIPAL LECTURES, Ryerson 251
Time	Mon, Mar 24	Tue, Mar 25	Wed, Mar 26	Thu, Mar 27	Fri, Mar 28
8:30 am
9:00 am		Zelmanov [0]	Turull [0]	Aschbacher [0]	Robinson
9:40 am		Leedham-Green [0]	Lucchini [0]	Chermak [0]	Srinivasan
10:20 am		Sidki [0]	Monti [0]	Oliver [0]	Kessar
10:50 am		Tea	Tea	Tea	[0]
11:20 am		Jaikin-Zapirain [0]	Khukhro [0]	Flavell [0]
12:00 pm		L. Wilson [0]	J.S. Wilson [0]	Stellmacher [0]
1:00 pm
2:10 pm	Sin [0]
2:50 pm	Solomon			SHORT TALKS
3:10 pm	Tea at 3:20pm	Isaacs [0]	SOFTBALL	STARTING 2:40pm
3:50 pm	Smith [0]	Nenciu [0]	(weather	PARALLEL
4:30 pm	Griess [0]	Alperin [0]	permitting)	SESSIONS

 

THURSDAY AFTERNOON SHORT TALK SCHEDULE
	Session A Ryerson 251	Session B Eckhart 133	Session C Eckhart 202
2:40	Geline [0]	Wilson [0]	Iverson [0]
3:10	Yong [0]	Russo [0]	Waldecker [0]
3:40	Bonner [0]	Morigi [0]	Guest [0]
4:10	Le [0]	Park [0]	Burness [0]
4:40	Adan-Bante [0]	Onofrei [0]	Fairbairn [0]

 

TITLES AND ABSTRACTS OF PRINCIPAL LECTURES

JONATHAN L. ALPERIN, The University of Chicago, "Split Extensions Revisited"
ABSTRACT: The relation between group extensions and ring extensions is discussed.


MICHAEL ASCHBACHER, Cal Tech, "The Local Theory Of Fusion Systems"
ABSTRACT: Fusion systems first arose in the context of modular representation theory, and in recent years have been studied intensively by homotopy theorists. I will discuss efforts to extend the local theory of finite groups to the category of saturated fusion systems. There is some hope that certain arguments are easier in the latter category, hence possibly leading to simplifications in the proof of the classification of the finite simple groups.

ANDREW CHERMAK, Kansas State University, "Mappings of p-Local Finite Groups"
ABSTRACT: In the category of saturated fusion systems, Aschbacher has introduced a notion of normal subsystem. In this category, it is not always the case that kernels of morphisms are normal subsystems. We propose a notion of morphism of p-local finite groups in which kernel fusion systems are normal.

PAUL FLAVELL, University of Birmingham, "A New Proof of the Solvable Signalizer Functor Theorem"
ABSTRACT: A new proof of Glauberman’s Solvable Signalizer Functor Theorem will be presented. The proof is based on Bender’s very short proof of the 2-Signalizer Functor Theorem. Previously, there had been considerable difficulties in extending Bender’s argument because Glauberman’s ZJ-Theorem breaks down at the prime 2. A new result on Primitive Pairs,related to the ZJ-Theorem and another result of Glauberman on Failure of Factorization overcomes this difficulty.

ROBERT L. GRIESS, Jr., The University of Michigan, "Those Darn Dihedral Groups"
ABSTRACT: In any group, a pair of involutions generates a dihedral group. Associated to a sublattice S of an integral lattice L is an involution (defined to be -1 on S and 1 on the annihilator of S ). This involution leaves L invariant if S satisfies the RSSD condition (2L is in S + ann(S )). If 2 * dual(S ) is contained in S , this holds trivially. For example, this holds when S is isometric to sqrt(2) * U, where U is integral and unimodular. A pair of RSSD sublattices gives a dihedral group in the isometry group of L.

In joint work, Ching Hung Lam and I have classified pairs M , N of lattices isometric to EE_8 := \sqrt(2) * E_8 such that the lattice M + N is integral and rootless and such that the dihedral group associated to them has order at most 12. Most of these pairs may be embedded in the Leech lattice. I shall discuss our classification in the lecture.

Our motive is to study the Glauberman-Norton theory about pairs of 2A involutions in the Monster and the extended Dynkin diagram for E_8. We shall analyze the Glauberman-Norton subgroups of the Monster by pairs of EE_8 sublattices of the Leech lattice and their associated subVOAs in the Moonshine VOA. One interesting point is that the Glauberman-Norton groups have "Levi factors" which look like index 2 subgroups of direct products of Weyl groups (mod centers). We hope to explain this in our VOA context.

I. MARTIN ISAACS, The University of Wisconsin, "The McKay Conjecture and Reduction to Simple Groups"
ABSTRACT: Given a finite group G and a prime number p, consider the number m(G) of irreducible characters of G that have degree not divisible by p. The McKay conjecture asserts that m(G) = m(N ) for every group G and every prime p, where N is the normalizer in G of a Sylow p-subgroup. This is known to be true for many types of groups, but despite its great generality, no one seems to understand why the conjecture should be true. So far, the best hope for a proof seems to be to ”reduce” the conjecture to a question about simple groups, and then to do a case-by-case check. This talk will discuss such a reduction.

ANDREI JAIKIN-ZAPIRAIN, Universidad Autonoma de Madrid, "On the Conjugacy Classes and Characters of Finite p-Groups"
ABSTRACT: In my talk I will present a survey of results about conjugacy classes and characters in finite p-groups.

RADHA KESSAR, University of Aberdeen, "Shintani Descent and Modular Gelfand-Graev Representations"
ABSTRACT: Given an extension of finite fields K/k, and f the corresponding Frobenius automorphism, Shintani descent provides a bijection between the set of f - stable ordinary irreducible characters of GLn (K) and the set of ordinary irreducible characters of GLn (k). (In the case that [K : k] and |GLn (k)| are relatively prime, this bijection is the Glauberman correspondence.) It is conjectured that under certain natural arithmetic conditions Shintani descent is the character-theoretic shadow of Morita equivalences between certain pairs of l-blocks of the two general linear groups, where l is a prime not dividing |k|. I will present an approach to this conjecture using Gelfand-Graev representations. This is joint work with C. Bonnafe'.


EVGENII KHUKRO, Cardiff University and Novosibirsk State University, "Large Characteristic Subgroups Satisfying Multilinear Commutator Identities"
ABSTRACT: Joint work with N. Yu. Makarenko. Some of these results were partly inspired by several recent papers of George Glauberman (some with J. Alperin) on the existence of normal abelian, or of small nilpotency class, subgroups of given index in finite p-groups. However, in our results, the exact index is sacrificed, being replaced by certain bounded index, in favor of more general conditions on the subgroup. If a group G has a nilpotent subgroup H of class c and of finite index n, then G has also a normal nilpotent subgroup H of class at most c and of index at most n!. But it is often required that this subgroup be normal in a larger group where G itself is a normal subgroup – for that we need a characteristic subgroup of G. Of course, the automorphic closure is a characteristic nilpotent subgroup of class at most nc. But in an induction on the length of a certain subnormal series, it may be crucial not to increase the nilpotency class at each step. We prove that G has a characteristic nilpotent subgroup of the same class at most c whose index is bounded in terms of n and c. Earlier such a result was known only for abelian subgroups. Moreover, we prove the same for arbitrary multilinear commutator identities (which define many other popular varieties: of soluble groups of derived length d, polynilpotent varieties, etc.) We also prove similar results for ideals in arbitrary (not necessarily associative or commutative or Lie or finite-dimensional) algebras. These results enabled us to finish the study of finite groups with "almost regular" automorphisms of order 4, where our results on Lie rings earlier were giving only certain subnormal series. We also obtain a result for locally nilpotent torsion-free groups, which improves the conclusion of our theorem on almost solubility of such groups admitting an almost regular automorphism of finite order. Finally, these results enabled the first author to finish the study of finite groups with an automorphism of prime order that is "almost regular" in the sense of the rank of the fixed-point subgroup.

CHARLES LEEDHAM-GREEN, Queen Mary & Westfield College, "The Classification of p-groups by coclass"

ANDREA LUCCHINI, Universita di Padova, "Sets of Elements that Pairwise Generate a Finite Group"
ABSTRACT: Let G be a 2-generated finite group. Define a (simple) graph Gamma(G) as follows: G is the vertex set and there is an edge between two vertices if and only if they generate G; denote by mu(G) the maximum size of a complete subgraph in Gamma(G). Recently mu(G) has been studied in connection with another invariant sigma(G) of G, namely, the least integer k such that G is the union of k of its proper subgroups. In general mu(G) is less than or equal to sigma(G), but for many non abelian simple groups, equality holds; moreover it has been conjectured that sigma(G)/mu(G) tends to 1 as |G| tends to infinity. This is false if G is not simple: for example we will show that if G is a 2-generated direct product of isomorphic non abelian simple groups, then sigma(G)/mu(G) tends to infinity. In general nothing is known about the behaviour of mu(G) for an arbitrary 2-generated group and the techniques that are typically employed in studying generation problems seems to be of no use. We will discuss some examples and propose questions and conjectures.

VALERIO MONTI, Universita di Roma "La Sapienza", "Pro-p-Groups with Few Normal Subgroups"
ABSTRACT: Motivated by the study of pro-p groups with finite coclass, we consider the class of pro-p groups with few normal subgroups. This is not a well defined class and we offer several different definitions and study the connections between them. Furthermore we propose a definition of periodicity for pro-p groups, thus providing a general framework for some periodic patterns that have been already observed in the existing literature. We then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups, but in addition have periodicity in the lattice of normal subgroups.


ADRIANA NENCIU, University of Wisconsin, "Brauer t-Tuples"
ABSTRACT: Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserves all the character values and the power map. Generalizing the definition of a Brauer pair we introduce thenotion of a Brauer t-tuple. We show that Brauer t-tuples exist for any t > 1.


BOB OLIVER, Universite de Paris 13, "Topological Tools for Studying Fusion Systems of Groups of Lie Type"
ABSTRACT: The talk will be centered around the following theorem, shown in joint work with Carles Broto and Jesper Moller. Fix a prime p, and let q and q’ be prime powers which are not powers of p, and such that they generate the same closed subgroup of Z_p^x . Then for any Lie group “type” G, the p-subgroup fusion systems of G(q) and G(q') are equivalent. In other words, the Sylow p-subgroups of G(q) and G(q') are isomorphic, via an isomorphism which preserves all fusion relations in G(q) and G(q') between their subgroups. A similar result holds for Steinberg groups.

This is a result which is not surprising to people who have studied these groups, but there does not yet seem to be a published proof. Our proof uses the connection between fusion and topology, taking as starting point a theorem of Martino and Priddy that two finite groups G and G' have equivalent fusion systems at p if their p-completed classifying spaces are homotopy equivalent. Other theorems, by Friedlander and Mislin, describe these classifying spaces as being "homotopy fixed point sets" of certain Frobenius endomorphisms. We show that under certain hypotheses on a space X, homotopy fixed point sets of two self equivalences of X are homotopy equivalent if they generate the same closed subgroup of the group of all self equivalences of X. Together with the other results described above, this proves the theorem, and also gives information about other equivalences between fusion systems of groups of Lie type.

GEOFFREY R. ROBINSON, University of Aberdeen, "Reductions (mod q) of amalgams at p"

SAID SIDKI, Universidade de Brasilia, "On Commutativity and Finiteness in Groups"
ABSTRACT: Let H, K be finite groups having equal orders n, and let f : H --> K be a bijection which maps the identity of H to the identity of K. We have studied various forms of weak permutability or weak commutativity of H and K. In particular, weak commutativity is formalized by the group

G(H, K; f ) =< H, K : h.h^f = h^f .h for all h in H > .

This group originated in connection with the following conjecture of 1976, which was proven recently by Aschbacher, Guralnick and Segev:

If a finite group G contains a non-trivial elementary abelian 2-group A such that every involution in G commutes with some involution from A, then A intersects O_2(G) non-trivially.

We will discuss in this lecture a number of results regarding the nature of G(H, K; f ). In case H and K are isomorphic to the elementary abelian p-group A_{p,k} of rank k, the notation is simplified to G(A_{p,k}, f). We will present results from joint work with R. Oliveira, which support the following conjecture:

The group G(A_{2,k}, f) has order dividing 2^{2^k + k - 1}, nilpotency class at most k + 2, and derived length at most k. For odd p, G(A_{p,k} , f) has order dividing p^{2k + (k(k-1))/2}, and nilpotency class at most 2.

PETER SIN, The University of Florida, "The Divisor Matrix, Dirichlet Series and S L(2, Z)"
ABSTRACT: We discuss actions on infinite dimensional rational vector spaces of some groups one of whose elements acts as the divisor matrix. In particular, we construct an integral representation of S L(2, Z ) on the space of sequences of rational numbers such that the coefficient sequences of convergent ordinary Dirichlet series form an invariant subspace on which the standard unipotent element acts as multiplication by the Riemann zeta function. Joint work with John G. Thompson.

STEPHEN D. SMITH, The University of Illinois at Chicago, "Glauberman's Work and the Classification of Simple Groups"
ABSTRACT: Glauberman’s work is prominent in many different areas of finite group theory; this talk will focus on applications in the classification of the finite simple groups. A deeper appreciation of this influence is emerging from the analysis in a joint project with Aschbacher, Solomon, and Lyons to give an outline of classification of groups of characteristic 2 type. For example, the Z*-Theorem became a basic tool in many different problems; it is even embedded in the design of the treatment of groups of GF(2) type. Another example of his influence is the circle of ideas called ”Glauberman’s Argument” in the Aschbacher-Smith quasithin project. There is no shortage of other examples.
 

RONALD SOLOMON, The Ohio State University, "George Glauberman's Impact Across Group Theory"

BHAMA SRINIVASAN, The University of Illinois at Chicago, "From Glauberman to Shintani: Applications to Representations of Finite Classical Groups"
ABSTRACT: The Glauberman correspondence for finite groups has many ramifications and is related to many other correspondences. In the case of finite groups of Lie type, it is analogous to the Shintani map which has applications to the representation theory of such groups. Some of these applications will be described. Finally, recent work on blocks of disconnected groups, where an analogous situation arises, will be discussed.

BERND STELLMACHER, Christian-Albrechts Universitat zu Kiel, "Finite Groups of Local Characteristic p with Large Subgroups"
ABSTRACT: Let G be a finite group. A p-subgroup Q is called large if C_G(Q) is contained in Q and N_G(U ) is contained in N_G(Q) for every non-identity subgroup U of Z(Q). Examples of finite groups possessing a large subgroup are the groups of Lie type in characteristic p, except for Sp(2n, 2^k), F_4(2^k) and G_2(3^k). In this talk, the p-local structure of finite groups of local characteristic p possessing a large subgroup will be discussed. In particular, it will be shown what kind of module-theoretic conditions and results are needed in this investigation.

ALEXANDRE TURULL, The University of Florida, "Above the Glauberman Correspondence"

ABSTRACT: The Glauberman correspondence is a fundamental tool for the study of the representations of finite groups. Let G be a finite group, p a prime, P a p-subgroup of G and N a normal p'-subgroup of G such that PN is also a normal subgroup of G. Let \psi be a P-invariant irreducible character of N, and \theta its Glauberman correspondent character, so that \theta is an irreducible character of C_N(P).  A theorem of Dade tells us that the characters of G lying above \psi are in one-to-one correspondence with the characters of N_G(P) lying above \theta, and that this correspondence has good compatibility properties. In fact, we can prove that there is, in a precise sense, an equivalence between the characters above \psi of subgroups of G that contain N and those above \theta of corresponding subgroups of N_G(P) that contain C_N(P) that preserves irreducible characters, induction, restriction, p-blocks, heights, Galois conjugation over the p-adics Q_p, and Schur indices over Q_p.  One consequence of this theorem is that the combined strengthenings of the McKay Conjecture due to Alperin, Isaacs, Navarro, and Turull holds for all p-solvable groups. This proof does not use the Classification of Finite Simple Groups.

JOHN S. WILSON, Oxford University, "Characterizations of solvability for finite groups and linear groups"

LARRY WILSON, Center for Communications Research, San Diego, "Groups with Fixed Point Free Automorphisms of Prime Order"
ABSTRACT: J. G. Thompson, in 1959, proved that a finite group with a fixed point free automorphism of prime order p must be nilpotent, settling a conjecture of Frobenius. The cases of p either 2 or 3 had long been known. In fact, if p = 2, then the group is Abelian and B.H. Neumann proved that, if p = 3, then the group is nilpotent of class at most 2. G. Higman, also in 1959, generalized these results by proving that, under the hypothesis of Thompson’s result, the nilpotency class is bounded by some function of p alone, now called h(p) in honor of Higman. Higman proved the lower bound that h(p) is at least (p^2 - 1)/4, but his proof did not provide an upper bound for h(p). Kreknin and Kostrikin were the first to give an explicit (exponential) upper bound for h(p). No improvements on the lower bound have been found, leading to the conjecture that h(p) = (p^2 - 1)/4. Some improvements on the upper bound have been found. We will discuss the history of this problem, including some recent results that may be the beginning of the proof of a polynomial upper bound for h(p).

EFIM ZELMANOV, The University of California at San Diego, "Lie Algebras of Infinite Differential Operators"

ABSTRACT: We will discuss a recently discovered class of Lie algebras that are close both to Grigorchuk groups and to certain physical constructions. There is a hope that these algebras will provide counterexamples to several old problems in ring theory.

 

 

EDITH ADAN-BANTE, Northern Illinois University, "Products and Restrictions of Characters"
ABSTRACT: Let G be a finite p-group, for some prime p, and \phi, \theta be irreducible complex characters of G. It has been proved that if, in addition, \phi and \theta are faithful characters, then the product \phi\theta is a multiple of an irreducible character or is the linear combination of at least (p+1)/2 distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k > 0, we can always find a p-group P and irreducible characters \phi and \theta of P such that the product \phi\theta is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.

TIM BONNER, The University of Florida, "Elements of the derived subgroup as products of commutators"
ABSTRACT: Let G be a finite nonabelian group. Every element of the derived subgroup G' is a product of commutators, but not necessarily a commutator. Let \lambda(G) be the smallest positive integer such that each element of G' can be written as a product of \lambda(G) commutators. We obtain new bounds on \lambda(G) in terms of the size of the derived subgroup, and answer a question in the Kourovka Notebook of unsolved problems in group theory. Namely, we prove that \lambda(G)/|G| is at most 1/6 for every finite group G, with equality exactly for S_3.

TIM BURNESS, University of Southampton, "Minimal generation of maximal subgroups of simple groups"
ABSTRACT: Let G be a finite group and let d(G) be the minimal number of generators for G. It is well-known that d(G) = 2 if G is a nonabelian finite simple group, while a theorem of Dalla Volta and Lucchini states that d(G) is at most 3 for any almost simple group G. Here we consider d(M), where M is a maximal subgroup of an almost simple group. In recent work with Liebeck and Shalev, we show that d(M) is at most 5. There are infinitely many examples with d(M) = 4, so this result is close to best possible. As an application, we deduce that if G is a finite primitive permutation group with point stabilizer H, then d(H) is at most d(G) + 5. In my talk, I will briefly discuss the main ideas and techniques used in the proof of these results.

BEN FAIRBAIRN, University of Birmingham, "Symmetric Presentations of Coxeter Groups"
ABSTRACT: The remarkable techniques of symmetric generation have furnished several almost effortless constructions of many exceptional objects by exhibiting highly symmetric generating sets for groups. In particular, these generating sets often provide symmetric presentations for these groups. In this short talk, we will show how the familiar Coxeter-Moser presentations for the Coxeter groups of types A_n, D_n, and even the exceptional cases E_6, E_7, and E_8, may be naturally derived as symmetric presentations in an almost uniform manner. In doing so, we will construct representations of these groups with some striking properties.

MICHAEL GELINE, The University of Chicago, "2-Local Schur Indices"
ABSTRACT: The Schur index is one of the most difficult invariants of an irreducible character to determine. The Schur index of \chi over the field F is the smallest multiplicity with which \chi can occur in a character afforded by an FG-module. It is clear that p-local representation theory should help in the case where F is a p-adic field. We present one phenomenon which is unique to the prime 2 and is related to Broue's abelian defect group conjecture.

SIMON GUEST, University of Southern California, "A Solvable Version of the Baer-Suzuki Theorem"
ABSTRACT: Let G be a finite group and x an element in G. Baer-Suzuki states that if < x,x^g > is nilpotent for every g in G, then < x^G > is nilpotent. If we replace the nilpotency condition with solvability, then there are counterexamples. For example, if x is an involution in a non-abelian simple group G, then < x,x^g > is dihedral, but < x^G > = G. Also, if x is a transvection in SL(n,3), then < x,x^g > is always solvable and again < x^G > = G. However, the following is true:

(1) If x has prime order p at least 5 and < x^G > is not solvable, then there exists g in G such that < x,x^g > is not solvable.

(2) If G is an almost simple group and x has odd prime order p, then one of the following holds:

(a) There exists g in G such that < x,x^g > is not solvable.

(b) F*(G) is a simple group of Lie type defined over F_3, or F*(G) is PSU(n,2), and the possible conjugacy classes of x are given explicitly. In particular, x has order 3.

As a corollary of (1) and (2), we have that if G is an arbitrary finite group and x has any order, then < x^G > is solvable if and only if for all g_1, g_2, and g_3 in G, < x,x^{g_1}, x^{g_2}, x^{g_3} > is solvable. The corollary will form part of a joint paper with Bob Guralnick and Paul Flavell, and has been announced independently by Plotkin, Gordeev, Grunewald and Kunyavskii. We will discuss these results, some recent improvements, and some of the methods used in the proof.

NATE IVERSON, Bowling Green State University, "A Phan-like theorem for orthogonal groups in even characteristic"
ABSTRACT: A new proof of Phan's theorem in 2001 gave a more general technique to identify groups via free amalgamated products. This will cover one such family of amalgams that can be produced for orthogonal groups in even characteristic and some low rank presentations.

TUNG LE, Wayne State University, "The Representations of the Upper Triangular Groups"
ABSTRACT: All the degrees of the irreducible characters of the upper triangular group U = U_n(q), q a power of a prime, are known. However the structure and representation of the irreducible characters are not fully understood. Here, we present a method for constructing and partitioning the irreducible characters of U. We call an irreducible U-module V almost faithful if the center Z(U) acts nontrivially on V. Each irreducible character is an irreducible constituent of tensor products of almost faithful characters \chi_i of pattern subgroups, which are isomorphic to U_{k_i}(q), k_i at most n. And an irreducible character uniquely determines the pattern subgroups and characters which are needed for its construction.

MARTA MORIGI, Universita di Bologna, "Generalizing a theorem of P. Hall on finite-by-nilpotent groups"
ABSTRACT: Let \gamma_i(G) and Z_i(G) denote the i-th terms of the lower and upper central series of a group G, respectively. P. Hall showed that if \gamma_{i+1}(G) is finite, then the index |G:Z_{2i}(G)| is finite. We prove that the same result holds under the weaker hypothesis that |\gamma_{i+1}(G): \gamma_{i+1}(G)\cap Z_i(G)| is finite.

SILVIA ONOFREI, Kansas State University, "On fixed point sets and Lefschetz modules for sporadic simple groups"
ABSTRACT: The reduced Lefschetz modules associated to complexes of distinguished p-subgroups (those subgroups which contain p-central elements in their centers) are investigated. The case when the underlying group G has parabolic characteristic p is analyzed in detail. We determine the nature of the fixed point sets of subgroups of order p. The p-central elements have contractible fixed point sets. Under certain hypotheses, the noncentral p-elements have fixed points which are equivariantly homotopy equivalent to the corresponding complex for a quotient of the centralizer. For the reduced Lefschetz modules, the vertices of the indecomposable summands and the distribution of these summands into the p-blocks of the group ring are related to the fixed point sets. Applications to the sporadic group geometries are discussed. This is joint work with J. Maginnis.

SEJONG PARK, The University of Aberdeen, "'Control of fusion' theorems of Glauberman and Thompson for fusion systems"
ABSTRACT: Fusion systems are categories satisfying essential features of fusion in finite groups. Motivated by Glauberman's 1969 Oxford Instructional Conference paper, "Global and local properties of finite groups", Kessar and Linckelmann have recently generalized Glauberman's ZJ-theorem and the Glauberman-Thompson p-nilpotency theorem to fusion systems. We follow Kessar and Linckelmann to generalize Thompson's p-nilpotency theorem to fusion systems via the path presented in Glauberman's paper (op.cit.), and as a consequence, we get a stronger version of Frobenius' p-nilpotency theorem for fusion systems when p is odd. This is joint work with Antonio Diaz, Adam Glesser and Nadia Mazza.

VALENTINA RUSSO, Universita dell'Aquila, "Centrally large subgroups of finite p-groups"

ABSTRACT: Let G be a finite p-group. A subgroup Q of G is centrally large, or a CL-subgroup of G, if |Q||Z(Q)| is greater than or equal to |Q*||Z(Q*)| for every subgroup Q* of G. For a CL-subgroup Q, |Q||C_G(Q)| is greater than or equal to |Q*||C_G(Q*)| for every subgroup Q* of G, and has several other properties. We study the self-minimal CL-groups, or SMCL-groups, G, i.e., the groups G such that G is a minimal CL-subgroup of itself, and we give some examples and counterexamples. We prove that an SMCL-group G has a proper subgroup H not equal to Z(G) such that |H||C_G(H)| = |G||Z(G)| if and only if G = HK, with H = C_G(K) and K = C_G(H). Moreover, if G has such a structure and H, K are SMCL-groups, then G is a CL-subgroup of itself, not necessarily minimal. The direct product of two SMCL-groups is an SMCL-group. We also show that the property of being an SMCL-group is invariant for isoclinism, hence it is not restrictive to consider the stem groups of the family of SMCL-groups, in order to classify this special class of p-groups.

REBECCA WALDECKER, University of Birmingham, "A local approach to Glauberman's Z*-Theorem"
ABSTRACT: We investigate a minimal counterexample to Glauberman's Z*-Theorem from a local group theoretic point of view. One of the results is a new proof in the special case where the centraliser of an isolated involution is soluble. However, the approach is general and we deduce information about the centraliser of an isolated involution, particularly about the components involved. We show how this affects the 2-rank of the group and give an overview of the most recent results.

JAMES B. WILSON, The University of Oregon, "Decomposing p-groups of class 2"
ABSTRACT: Central products of p-groups of class 2 are studied using Jordan and *-algebras. These algebras are used to prove that there are at least p^{2n^3/27 + Cn^2} centrally indecomposable p-groups of order p^n. A Krull-Remak-Schmidt type theorem is proved for central products of p-groups of class 2. Unlike that classical theorem for direct products, there are families of p-groups in which the action of the automorphism group on the set of fully refined central decompositions is intransitive: it can have an arbitrary number of orbits.

YANG YONG, The University of Florida, "Orbits of the actions of finite solvable groups"
ABSTRACT: Let G be a finite solvable irreducible linear group on a finite vector space V. It is not always true that there exists some v in V whose centralizer in G is trivial. Results of Moreto and Wolf show, however, that there will always exist some v in V whose centralizer is contained in the 9th Fitting subgroup of G. We show that in fact there will always exist some v in V whose centralizer in G is contained in the 7th Fitting subgroup of G, and that taking the 6th Fitting subgroup is not possible in general. This result has many applications. For example, as a corollary one obtains that if G is any solvable group, then G has a conjugacy class whose size is divisible by the index in G of the 8th Fitting subgroup of G.