相关论文: Arbitrarily large finite quotients imply no unifor…

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative…

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…

Let X be a finite CW complex. We show that the fundamental group of X is large if and only if there is a finite cover Y of X and a sequence of finite abelian covers \{Y_N\} of Y which satisfy b_1(Y_N)\geq N. We give some applications of…

We say a group is finitely annihilated if it is the set-theoretic union of all its proper normal finite index subgroups. We investigate this new property, and observe that it is independent of several other well known group properties. For…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…

We give an example of a finitely presented simple group containing a finitely generated subgroup which is not finitely presented.

We present two results on expansion of Cayley graphs. The first result settles a conjecture made by DeVos and Mohar. Specifically, we prove that for any positive constant $c$ there exists a finite connected subset $A$ of the Cayley graph of…

It is well-known that random groups with at least as many relators as generators have vanishing first betti number with high probability. In this paper we extend this question and study maps from few-relators random groups to infinite…

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect…

We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first…

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic…

Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of…

We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit…

A finitely generated solvable group with unbounded iterated identity is constructed.

We indicate a natural generalization of the concept of subgroup commutativity degree of a finite group and a list of open problems on these new concepts.

We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete…