The so-called Burnside-Dixon-Schneider (BDS) method currently used as the default method of computing character tables in GAP for groups which are not solvable is often inefficient in dealing with groups with large centres. If G is a finite group with centre Z and λ a linear character of Z, then we describe a method of computing the set Irr(G,λ) of irreducible characters χ of G whose restrictio...

Subgroups such as Hall systems, system normalizers, F-normalizers and F-covering subgroups arise naturally in the study of finite solvable groups. Here we present practical algorithms to compute such subgroups. Besides being of interest in the theory of formations, the algorithms also yield fast methods for constructing head complements. These complements are closely related to the maximal subg...

Let K be a noncommutative division ring with center Z and multiplicative group K*. Hua [2; 3] proved that (i) K*/Z* is a group without center, and (ii) K* is not solvable. A generalization (Theorem 1) will be given here which contains as a special case (Theorem 2) the fact that K*/Z* has no Abelian normal subgroups. This latter theorem obviously contains both (i) and (ii). As a further corollar...

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit ar...

Let G = 〈S〉 be a solvable permutation group given as input by the generating set S. I.e. G is a solvable subgroup of the symmetric group Sn. We give a deterministic polynomial-time algorithm that computes an expanding generating set of size Õ(n2) for G. More precisely, given a λ < 1, we can compute a subset T ⊂ G of size Õ(n2) ( 1 λ )O(1) such that the undirected Cayley graph Cay(G,T ) is a λ-s...

The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable nondegenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theo...

We associate a graph CG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete...

A group of order n is said to be R-sequenceable if the nonidentity elements of the group can be listed in a sequence a,, a,, . . . , a,_ 1 such that the quotients a; ‘az, a; ’ a3, . , a;?,a,_ r, a.-_‘,a, are distinct. An abelian group is R*-sequenceable if it has an R-sequencing a,, a,, . . , a,, _ 1 such that a,_ ra, + r = ai for some i (subscripts are read modulo n 1). Friedlander, Gordon and...

A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3-manifolds to the more ge...