Showing papers in "Journal of Group Theory in 2002"

Black-box recognition of finite simple groups of Lie type by statistics of element orders

Abstract: Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number of (nearly) uniformly distributed random elements of G, and examines the divisibility of the orders of these elements by certain primitive prime divisors. We show that the divisibility statistics determine G, except that we cannot distinguish the groups PWð2mþ 1; qÞ and PSpð2m; qÞ in this manner when q is odd and md 3. These two groups can, however, be distinguished by using an algorithm of Altseimer and

On the power structure of powerful p-groups

Abstract: Abstract We prove that the exponent Ω m (G) is at most p m if G is a powerful p-group with p odd. Calling on a recent result of Héthelyi and Lévai, we prove that |G p m | = |G:Ω m (G)| for all m. These results also hold for regular p-groups. We also bound the nilpotence class of a subgroup of a powerful group by e + 1, where p e is the exponent of the subgroup. This is just one more than what the bound would be if the subgroup were itself powerful.

Classifying spaces for proper actions of locally-finite groups

Abstract: For each finite ordinal n and each locally-finite group G of cardinality aleph n, we construct an (n+1)-dimensional model for the classifying space for proper G-actions. We use this complex to obtain information about the cohomology of G with free coefficients. Our techniques also give information about the location of large free abelian groups within Kropholler's hierarchy.

Groups generated by two bicyclic units in integral group rings

Abstract: The first author has been partially supported by the Onderzoeksraad of Vrije Universiteit Brussel and the Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) and the second by the D.G.I. of Spain and Fundacion Seneca of Murcia.We would like to express our gratitude to Victor Jimenez for some helpful conversation on inequality .

Subgroups of constructive nilpotent-by-abelian groups and a generalization of a result of Bieri, Neumann and Strebel

Abstract: Abstract We prove a ∑-version of the result of Bieri, Neumann and Strebel [R. Bieri, W. D. Neumann and R. Strebel. A geometric invariant of discrete groups. Invent. Math. 90 (1987), 451–477] that for a finitely presented group G without free subgroups of rank 2 the set ∑1(G) c has no antipodal points. More precisely, we prove that for such a group G We show that if G is a finitely generated nilpotent-by-abelian group then The latter result is used in constructing a counter-example to a conjecture of Meinert [H. Meinert. Iterated HNN-decomposition of constructible nilpotent-by-abelian groups. Comm. Algebra 23 (1995), 3155–3164] concerning the homological properties of subgroups of constructible nilpotent-by-abelian groups.

On pointwise conjugacy of distinguished coset representatives in Coexeter groups

Abstract: Let (W, S) be a Coxeter system. For a standard parabolic. subgroup W-K, K subset of or equal to S let D-K be the set of distinguished coset representatives, i.e. representatives of cosets W(K)w of minimal Coxeter length. If L = K-c subset of or equal to S with c is an element of W, then D-K and D-L = c(-1) D-K are in general not conjugate as sets. However it is shown that if WK is finite, they are conjugate 'pointwise', i.e. there is a bijection theta : D-K --> D-L such that theta(d) = d(wc) for some w is an element of W-K depending on d is an element of D-K. In particular for each conjugacy class C of W the cardinalities # (D-K boolean AND C) and # (D-L boolean AND C) are the same. The case of infinite standard parabolic subgroups is also discussed and a corresponding result is proved.

The geometry of groups satisfying weak almost-convexity or weak geodesic-combability conditions

Abstract: We examine the geometry of the word problem of two different types groups: those satisfying weak almost-convexity conditions and those admitting geodesic combings whose width satisfy minimally restrictive, non-vacuous constraints. In both cases we obtain an n! isoperimetric function and n 2 upper bounds on the minimal isodiametric function and the filling length function. 2000 Mathematics Subject Classification: 20F05, 20F06, 20M65

On the uniqueness of factors of amalgamated products

Abstract: Let G = A ∗C B be an amalgamated product. We study conditions under which the factor B is determined by A and C, i.e. under which the existence of another splitting of G as an amalgamated product G = A ∗C D implies B = D or B ∼= D. We also describe the structure of the family of amalgamated products along a given malnormal subgroup. There are situations where D 6∼= B. We describe them and show that they are the only ones after giving a short account of controlled subgroups as introduced in [FRW]. Mary Jones has independently constructed an example of the type below and Ilya Kapovich has provided an insightful example to a related question. This work was motivated by the first author’s work on splittings of surface groups [B].

Groups Whose Universal Theory Is Axiomatizable by Quasi-Identities

Abstract: Discriminating groups were introduced in [3] with an eye toward applications to the universal theory of various groups. In [6] it was shown that if G is any discriminating group, then the universal theory of G coincides with that of its direct square G G. In this paper we explore groups G whose universal theory coincides with that of their direct square. These are called square-like groups. We show that the class of square-like groups is first-order axiomatizable and contains the class of discriminating groups as a proper subclass. Further we show that the class of discriminating groups is not first-order axiomatizable.

Nilpotent groups with every quotient residually finite

Abstract: Abstract Let G be a nilpotent group with torsion subgroup τ(G). Then every quotient of G is residually finite if and only if G/τ(G) has no quasicyclic section and every primary component of τ(G) is an abelian-by-finite group with finite exponent.

An invariant of elements of finite order in semisimple simply connected algebraic groups

Abstract: This paper is a continuation of the paper [14], henceforth cited as I. Let k be a field of characteristic $ and ks a separable closure of k. Let G/k be a semisimple simply connected algebraic group which is assumed absolutely almost k-simple, i.e. G × kks is almost simple. Let $ be an element of order n. The goal of the paper is to define and study an invariant $ with value in K2(k)/nK2(k), where K2(k) denotes the second Milnor K-group of k. This invariant is characteristic-free and therefore it permits us to work in an arithmetical set-up. The case when n = p = char(k) > 0 and G is split is especially interesting, and an element of order p is then unipotent. In [28], Tits constructed explicit examples of anisotropic unipotent elements, i.e. unipotent elements which do not belong to any proper k-parabolic subgroup of G, and a large part of this paper is devoted to the computation of the invariant for such elements. For the split groups of type G2. (resp. F4, E8) and n =p = 2 (resp. 3, 5), we show that this invariant classifies conjugacy classes of anisotropic elements of order p, and it turns out that any such element is conjugate to one of the anisotropic elements constructed by Tits. In particular, we prove the remaining case of Theorem 1.3, i.e. that any anisotropic unipotent element of order 5 in the split group E8/k normalizes a maximal k-split torus.

The principal block of ℤ p S 2p

Abstract: Let p be an odd prime. This note describes the principal block $ of the group ring of the symmetric group S2p over the p-adic integers $. The principal block $ of $ has been considered in several papers: [4] determines the Ext-quiver of $ [5], the Loewy series of the projective indecomposable $-modules and finally [1] gives a presentation for the basic algebra that is Morita equivalent to $, where some constants are not fully determined. The present note does not depend on these results. The most important starting point used here is the decomposition matrix of $, which is calculated with the JantzenSchaper formula [11], [3]. Then the description of $ is a very easy application of the general theory developed by Plesken [8], [9] and further in [7]. Scopes [12] shows that many properties of $ also hold for all other symmetric group blocks of defect 2. We are confident that the same methods as those used here can be applied to describe these blocks up to Morita equivalence.

On the centers of maximal subgroups of locally finite simple groups of alternating type

Abstract: Let G be a locally finite simple group of alternating type, p a prime, and Z ≤ G be elementary abelian of order p. We prove that there exists 1 6= z ∈ Z with CG(z) 6= CG(Z).

Fischer subgroups, Fitting height, and pronormality

Abstract: Several authors have studied the relationship betweeen Fischer $-subgroups and $injectors of a finite soluble group. Fischer [6] proved that when $ is a special kind of Fitting set, now called a Fischer set of G, the Fischer $-subgroups and the $-injectors of G coincide. A bit more generally, Anderson [1] came to the same conclusion when the Fischer $-subgroups of H are normally embedded in H for each subgroup H of G. In [4] and [5] respectively, the second author proved the same thing when the Fischer $-subgroups of G are either subnormally embedded or locally pronormal in G, and when they are system permutable. The first author [2] provided the original example of a Fischer $-subgroup that is not an $-injector, where $ is a Fitting class; later Doerk and Hawkes [3, VIII(4.9)] gave an example of a group G and a Fitting set $ such that the Fischer $-subgroups of G are not injectors for $ or for any other Fitting set of G. In this paper, we provide one more result like those above, and construct two counterexamples. First, we prove that if $ is a Fitting set and a Fischer $-subgroup of a finite soluble group is nilpotent, then it must be an $-injector of that group. Next, we give an example of a finite soluble group with a Fitting set $ and a Fischer $-subgroup of Fitting height 2 (isomorphic to the symmetric group of order 6) that is not an $-injector; this example is substantially less complicated than that of [2]. Finally, answering a question of Doerk and Hawkes [3, p. 562], we construct a finite soluble group G possessing a Fitting set $ with a Fischer $-subgroup that is not pronormal in G. Because injectors are pronormal, this example generalizes that of [3, VIII(4.9)], and indeed G has a subgroup G1 of index 2 which contains all the subgroups in $, such that G1 and $ are similar to the group and the Fitting set constructed by Doerk and Hawkes. Most of our notation, like $, Syl3(G), Ker τ, O5(G) and O{3, 5}(G) is the same as that used by Doerk and Hawkes [3]. If $ is a Fitting set of G and $, we denote by $ the Fitting set inherited by H, but we drop the subscript unless it is required for clarity. For $, let