Volume 11, Issue 6

We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralizers in G . If G is a finite simple group of Lie type in odd characteristic, then this reduction can be performed in (Monte Carlo) polynomial time.

The product replacement algorithm is a practical algorithm to construct random elements of a finite group G . It can be described as a random walk on a graph Г k ( G ) whose vertices are the generating k -tuples of G (for a fixed k ). We show that if G = PSL(2, q ) or PGL(2, q ), where q is a prime power, then Г k ( G ) is connected for any k 4. This generalizes former results obtained by Gilman and Evans.

We apply a method of Bender [H. Bender. Finite groups with large subgroups. Illinois J. Math . 18 (1974), 223–228.] to determine the order of a group G of J 3 -type. Moreover, we determine the local p -structure of G for every prime p dividing the order of G . The results of this paper are obtained by exploiting the action of G on its geometry [B. Baumeister. A geometry for groups of J 3 -type. Arch. Math. ( Basel ) 88 (2007), 491–499.] and by sophisticated use of elementary group theory.

We show that a connected split reductive group G over a field of characteristic 0 is determined up to isomorphism by specifying a maximal torus T of G , the set of isomorphism classes of irreducible representations of G , and the character homomorphism from the Grothendieck ring of G to that of T . More precisely, we determine all isomorphisms compatible with the specified data.

We know that almost all Brauer trees are shaped like a star. Given such a star and an odd prime l , we give an explicit method for constructing infinitely many groups with this star as the Brauer tree of some l -block. Furthermore we show that there is an infinite family of Brauer trees which cannot be realized in the principal block of any group.

In this paper, we prove that each component of the Burnside ring of a finite group is the soluble component of the Burnside ring of a Weyl subgroup of its corresponding group. We show that groups with isomorphic Burnside rings have the same sublattice of soluble normal subgroups and the same spectrum. This gives for the alternating groups, the sporadic simple groups and many series of simple groups of Lie type a positive answer to Yoshida's question when a finite group is determined by its Burnside ring up to isomorphism.

Let G be a permutation group acting on a finite set Ω. An uncovering-by-bases (or UBB) for G is a set of bases for G such that any r -subset of Ω is disjoint from at least one base in , where , for d the minimum degree of G . The single-orbit conjecture asserts that for any finite permutation group G , there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k -transitive and has a base of size k + 1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size.

For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈 a, b 〉 ∈ ℱ for all a ∈ A and b ∈ B . This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [ A, B ] ⩽ F ( G ), where F ( G ) denotes the Fitting subgroup of G . Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.

We define a reflection in a tree to be an involutive automorphism whose set of fixed points is a geodesic, and we prove that, for the case of a homogeneous tree of even degree, the group of even automorphisms may be covered by at most seven reflections.

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag–Solitar group BS( p, q ) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag–Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product.

A new bound for the rank of the intersection of finitely generated subgroups of a free group is given, formulated in topological terms, and in the spirit of Stallings [J. R. Stallings. Topology of finite graphs. Invent. Math . 71 (1983), 551–565.]. The bound is a contribution to the strengthened Hanna Neumann conjecture.