Related papers: Algorithms for the Tits alternative and related pr…

Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$.…

Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In…

Building upon the author's previous work on primitivity testing of finite nilpotent linear groups over fields of characteristic zero, we describe precisely those finite nilpotent groups which arise as primitive linear groups over a given…

We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of…

Motivated by the need for efficient isomorphism tests for finite groups, we present a polynomial-time method for deciding isomorphism within a class of groups that is well-suited to studying local properties of general finite groups. We…

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Hard instances of natural computational problems are often elusive. In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find. More…

We prove the Tits Alternative for groups acting on $2$-dimensional $\mathrm{CAT}(0)$ complexes with a bound on the order of the cell stabilisers.

We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…

We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming…

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence…

Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form…

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by…

In this survey we show how well known results about the Word Problem for finite group presentations can be generalized to the Word Problem and other decision problems for non-necessarily finite monoid and group presentations. This is done…

We study the problem of deciding universal termination of linear and affine loops over the reals in the bit-model of real computation. We show that both problems are as close to decidable as one can expect them to be: there exist sound…

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

In this note we prove the following results: $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT, then $G$ has decidable word problem. More generally, for f.g. groups that are not recursively presented, there exists…

We construct a finitely presented group with undecidable word problem and with Dehn function bounded by a quadratic function on an infinite set of positive integers.

We show that the epimorphism problem is solvable for targets that are virtually cyclic or a product of an Abelian group and a finite group.

Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path to proof by induction on the order of a group, which is required for most interesting…