Results on the equivalence problem for finite groups

The equivalence problem for a finite nilpotent group has polynomial time complexity, even when the terms have parameters from the group. The same result holds for the dihedral groups D_n.Received August 24, 2002; accepted in final form August 5, 2004.     

A GENERALIZATION OF COMPLETE GROUPS

Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σ _n , where σ _n is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel the results found in the author's dissertation for Lie algebras. There it is shown that only complete Lie algebras have the desired property. Thus, these results provide an example of when the theory of Lie algebras diverges from that of groups.     

POWER GROUPS OF FREE ABELIAN GROUPS OF FINITE RANK

ABSTRACT

The power set of a group G has an induced semigroup structure, some subsets of which will form groups in their own right. We are especially interested in such subsets that are maximal. We demonstrate that even when G is a free abelian group of finite rank, the groups which arise in this way can be diverse profinite abelian groups.     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ?(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.     

Structure of Quantum Doubles of Finite Clifford Monoids

Abstract

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

Kazhdan constants for conjugacy classes of compact groups

Kazhdan constants relative to conjugacy classes of compact groups are computed. They depend on the nontrivial irreducible characters of the respective group. The result is applied, in particular, to finite groups of Lie type, symmetric groups, and the group SU(n).     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

Groups of binary relations

It was shown in [3] that every finite group is the maximal subgroup of a semigroupB _x of all binary relations on some finite set X. This result is extended here to arbitrary groups.     

A Generalized Hughes Property of Finite Groups

Abstract

A Hughes cover for exponent p(pa prime number) of a finite group is a union of subgroups whose (non-empty) complement consists of elements of order p. A proper Hughes subgroup is an instance of a Hughes cover; and its parent group is soluble by a well-known result of Hughes and Thompson. More generally an earlier result of the authors shows that a group with a Hughes cover of fewer than psubgroups is soluble. This article treats the insoluble groups having a Hughes cover for exponent pwith exactly psubgroups: the almost simple groups with this property form a restricted class of projective special linear groups.     

Weak containment in the space of actions of a free group

It is shown that the translation action of the free group with n generators on its profinite completion is the maximum, in the sense of weak containment, measure preserving action of this group. Using also a result of Abért-Nikolov this is used to give a new proof of Gaboriau’s theorem that the cost of this group is equal to n. A similar maximality result is proved for generalized shift actions. Finally a study is initiated of the class of residually finite, countable groups for which the finite actions are dense in the space of measure preserving actions.     

The classification of flag-transitive Steiner 4-designs

Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades flag-transitive Steinert-designs (i.e. flag-transitive t-(v,k,1) designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple group classification, for Steiner t-designs with parameters t > 2 such characterizations have remained challenging open problems for about 40 years (cf. [11, p. 147] and [12 p. 273], but presumably dating back to around 1965). The object of the present paper is to give a complete classification of all flag-transitive Steiner 4-designs. Our result relies on the classification of the finite doubly transitive permutation groups and is a continuation of the author's work [20, 21] on the classification of all flag-transitive Steiner 3-designs. 2000 Mathematics Subject Classification. Primary 51E10 . Secondary 05B05 . 20B25     

ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS

Abstract

This paper is devoted to the study of groups with the property that the Frattini factor group is a T-group, i.e. a group in which every subnormal subgroup is normal. We give necessary and suffucient conditions for a direct product G = H x K of finite groups H and K to have such a property. Some structure theorems are also discussed.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

A rigidity theorem for automorphism groups of trees

A group G is called unsplittable if Hom(G, ℤ) = 0 and this group is not a non-trivial amalgam. Let X be a tree with a countable number of edges incident at each vertex and G be its automorphism group. In this paper we prove that the vertex stabilizers are unsplittable groups. Bass and Lubotzky proved (see [3]) that for certain locally finite trees X, the automorphism group determines the tree X (that is, knowing the automorphism group we can “construct” the tree X). We generalize this Theorem of Bass and Lubotzky, using the above result. In particular we show that the Theorem holds even for trees which are not locally finite. Moreover, we prove that the permutation group of an infinite countable set is unsplittable and the infinite (or finite) cartesian product of unsplittable groups is an unsplittable group as well. This research was supported by the European Social Fund and National resources-EPEAEK II grant Pythagoras 70/3/7298.