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Alexey Muranov

Publications and source records attributed to Alexey Muranov.

9 recordsLinked to original sources
arXiv2022

Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms

If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of $\operatorname{End}_R(M)$ determines a left $\operatorname{End}_{R}(M)[X]$-module structure on $\operatorname{End}_{R}(M)$, where $\operatorname{End}_{R}(M)$ is the $R$-algebra of endomorphisms of $M$ and $\operatorname{End}_{R}(M)[X] =\operatorname{End}_{R}(M)\otimes_RR[X]$. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method.

math.HO↗
arXiv2010

A remark on groups without finite quotients

We notice that the class of nontrivial groups without proper subgroups of finite index is not elementary, because some groups in this class, such as $\mathbb Q*\mathbb Q$, have ultrapowers that map homomorphically onto $\mathbb Z/p\mathbb Z$ for every prime $p$. Also, some ultrapowers of certain simple groups map homomorphically onto $\mathbb Z/2\mathbb Z$.

math.GR↗
arXiv2009

Interpretation of the Arithmetic in certain groups of piecewise affine permutations of an interval

The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.

math.GR↗
arXiv2009

Interprétation de l'Arithmétique dans certains groupes de permutations affines par morceaux d'un intervalle

The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.

math.LO↗
arXiv2009

Finitely generated infinite simple groups of infinite commutator width

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

math.GR↗
arXiv2007

Independence property and hyperbolic groups

We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.

math.LO↗
arXiv2007

On torsion-free groups with finite regular file bases

The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group $G$ has a finite system of generators $a_1$, ..., $a_n$ such that every element of $G$ has a unique presentation in the form $a_1^{k_1}... a_n^{k_n}$ where $k_i\in\mathbb Z$, is it true that $G$ is virtually polycyclic? The answer is ``not always.'' A counterexample is constructed in this paper as a group presented by generators and defining relations.

math.GR↗