Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

Latin trades on three or four rows

Latin trades are closely related to the problem of critical sets in Latin squares. We denote the cardinality of the smallest critical set in any Latin square of order n by scs(n). A consideration of Latin trades which consist of just two columns, two rows, or two elements establishes that scs(n)?n-1. We conjecture that a consideration of Latin trades on four rows may establish that scs(n)?2n-4. We look at various attempts to prove a conjecture of Cavenagh about such trades. The conjecture is proven computationally for values of n less than or equal to 9. In particular, we look at Latin squares based on the group table of Z_n for small n and trades in three consecutive rows of such Latin squares.     

Uniquely Edge-3-Colorable Graphs and Snarks

 A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|). Received: July 14, 1997 Revised: June 11, 1998     

All group‐based latin squares possess near transversals

In a latin square of order n, a near transversal is a collection of n ?1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.     

Irreducible Restrictions of Representations of the Symmetric Groups

A proof is given of a recent conjecture of Jantzen and Seitzgiving a necessary and sufficient condition for a representationof the symmetric group on n objects (over an algebraically closedfield of prime characteristic p < n) to remain irreducibleupon restriction to the symmetric group on n–1 objects.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Partial Latin Squares Are Avoidable

A square array is avoidable if for each set of n symbols there is an n × n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m ≥ 2 any partial Latin square of order 4m − 1 is avoidable, thus concluding the proof that any partial Latin square of order at least 4 is avoidable.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K _n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product of graphs. As the main result of this paper, we prove that for any two graphs G ₁ and G ₂ with η(G ₁) = h and η(G ₂) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following:

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

Barely Transitive and Heineken Mohamed Groups

A negative answer to the Kuro–ernikov Question 21 in [7],whether a group satisfying the normalizer condition is hypercentral,was given by Heineken and Mohamed in 1968 [6]. They constructedgroups G satisfying: (i) G is a locally finite p-group for a prime p, (ii) G/G'C_p and G' is countable elementary abelian, (iii) every proper subgroup of G is subnormal and nilpotent, (iv) Z(G)={1}, (v) the set of normal subgroups of G contained in G' is linearlyordered by set inclusion, see [3, p. 334], (vi) KG' is a proper subgroup in G for every proper subgroupK of G, see [6, Lemma 1(a)].     

A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w(x₁, ...,x_n) be a group word. Then the probability that w(g₁, ..., g_n)= 1 (where (g₁, ..., g_n) is a random n-tuple in G) is at leastp^–(m–t), where p is the largest prime divisor ofm and t is the number of distinct primes dividing m. This contrastswith the case of a non-soluble group G, for which Abérthas shown that the corresponding probability can take arbitrarilysmall positive values as n .     

Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d–1)/d and the minimum is taken over all criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)     

On the direct product conjecture for sigma invariants

We show that the direct product conjecture for ⁿ(G; ), whereG is the direct product of two groups of type FP_n, holds forn = 3 and give counterexamples for n 4. Previously, counter-exampleswere known only for a related conjecture involving the homotopical-invariants, where the conjecture already fails for n 3.     

Irreducible subrepresentations of the conjugation representation of finitep-groups

For finitep-groupsG we study the conjugation representation γG which is defined by lettingG act on itself by conjugation. Roth conjectured that every irreducible representation ofG which is trivial on the center ofG, occurs in γG. However, this is not true in general. We construct minimal counterexamples and verify Roth's conjecture for various classes of finitep-groups, for instance those of maximal class. Dedicated to Professor George Maltese on the occasion of his 60th birthday     

Chracters on Weighted Amenable Groups

In this paper we prove that for every weight on an amenablegroup there is always a continuous bounded character on thatgroup. Thus we may assume that any weight on an amenable groupis always greater than 1. Using a result of N. Grønbæk[1], this implies that the only amenable weighted group algebrasare up to isomorphism L¹(G)for some amenable group G. A Hahn–Banachtype generalisation is given for the extension of bounded charactersand examples are given showing that the assumption of amenabilityis necessary.     

Dynamique des Varietes Affines

An n-affine manifold is a differentiable manifold endowed withan atlas whose transition functions are locally affine transformationsof Rⁿ. The paper studies affine dynamic, that is, affine endomorphismsof affine manifolds. The main goal is to relate the dynamicalviewpoint to the completeness problem. In particular, it isshown that the Markus conjecture is true when dim Aff(M, ) >dim M – 2.     

On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.