Related papers: The Finitary Andrews-Curtis Conjecture
The paper discusses the Andrews-Curtis graph of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one them can be obtained from…
The Andrews-Curtis conjecture remains one of the outstanding open problems in combinatorial group theory. It claims that every normally generating $r$-tuple of a free group $F_r$ of rank $r\geq 2$ can be reduced to a basis by means of…
The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ by an arbitrary…
The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary…
It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in…
The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with…
Recently, sub-indices and sub-factors of groups with connections to number theory, additive combinatorics, and factorization of groups have been introduced and studied. Since all group subsets are considered in the theory and there are many…
In this paper we formulate and prove a combinatorial version of the section conjecture for finite groups acting on finite graphs. We apply this result to the study of rational points and show that finite descent is the only obstruction to…
In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR. First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete…
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
We show that the Andrews-Curtis conjecture holds for all balanced presentations of the trivial group corresponding to Heegaard diagrams of $S^3$.
We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional…
We define a class of finite groups based on the properties of the closed twins of their power graphs and study the structure of those groups. As a byproduct, we obtain results about finite groups admitting a partition by cyclic subgroups.
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…
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
We generalize the Alvis-Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis-Curtis duality of infinite type which…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained.