arXiv subjects
Publications and source records attributed to Anton A. Klyachko.
Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation $x^n=1$ in a group, the Solomon theorem (1969) on the number of solutions in a group to a system of equations having fewer equations than unknowns, and the Iwasaki theorem (1985) on roots of subgroups. There are other curious corollaries on groups and rings.
This note contains a (short) proof of the following generalisation of the Friedman--Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $rank(A\cap B)-1\leqslant n\cdot(rank(A)-1)\cdot(rank(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.
Any virtually free group $H$ containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing $H$ as a verbally closed subgroup.
It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.
We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative ring, the number of Pythagorean triples (as well as four-tuples, etc.) of invertible elements is a multiple of the order of the multiplicative group.
Generalising Solomon's theorem, C. Gordon and F. Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of i-th unknown in j-th equation is less than the number of unknowns. We generalise this result in two directions: first, we consider equations with coefficients, and secondly, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order this subgroup.
Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.
We prove that, in any field of characteristic not two and not three except the five-element field, each element decomposes into a product of four factors whose sum vanishes. We also find all $k,n,q$ such that every $n\times n$ matrix over the $q$-element field decomposes into a product of $k$ commuting matrices whose sum vanishes.
We prove a generalisation of the Khukhro--Makarenko theorem on large characteristic subgroups with laws. This general fact implies new results on groups, algebras, and even graphs and other structures. Concerning groups, we obtain, e.g., a fact in a sense dual to the Khukhro--Makarenko theorem. A graph-theoretic corollary is an analogue of this theorem in which planarity plays the role of a multilinear identity. We answer also a question of Makarenko and Shumyatsky.
We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in the sense that $G$ contains an infinite abelian normal subgroup while all finite Cartesian powers of $G$ are algorithmically finite (i.e., for any positive integer $n$, there is no algorithm which writes out an infinite sequence of pairwise different elements of $G^n$). We also state several related problems.
A nontrivial commutator cannot be a proper power in a torsion-free group satisfying C'(λ) small cancellation condition with sufficiently small λ.
Operations of arbitrary arity expressible via addition modulo 2^n and bitwise addition modulo 2 admit a simple description. The identities connecting these two additions have finite basis. Moreover, the universal algebra with these two operations is rationally equivalent to a nilpotent ring and, therefore, generates a Specht variety.
A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.
How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.
If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word "large", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups.
An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.
Suppose that G is a nontrivial torsion-free group and w is a word in the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from w by erasing all letters belonging to G is not a proper power in the free group F(x_1,...,x_n). We show how to reduce the study of the relative presentation Ĝ= to the case n=1. It turns out that an "n-variable" group Ĝ can be constructed from similar "one-variable" groups using an explicit construction similar to wreath product. As an illustration, we prove that, for n>1, the centre of Ĝ is always trivial. For n=1, the centre of Ĝ is also almost always trivial; there are several exceptions, and all of them are known.
We give a short proof and some strengthening of the Khukhro--Makarenko theorem that each group virtually satisfying an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained.