# Ilya B. GorskovSobolev Institute of Mathematics, Russian Academy of Sciences · Algebra

Ilya B. Gorskov

PhD

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60

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Introduction

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## Publications

Publications (60)

Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peir...

If G is a finite group, then the spectrum ω(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (G)$$\end{document} is the set of all element orders of G. The prim...

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called [Formula: see text]-axial algebras, where [Formula: see text] is an elemen...

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if $\mathbb{F}$ is a field of characteristic 0, then there exist infinitely many primitive axial algebras of Jordan type $\frac{1}{2}$ over $\mathbb{F}$ that are not factors of Matsuo algebras....

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called $\mathcal{PC}(\eta)$-axial algebras, where $\eta$ is an element of the gro...

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if F is a field of characteristic 0, then there exist infinitely many primitive axial algebras of Jordan type 1/2 over F that are not factors of Matsuo algebras. As an illustrative example, we...

The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer k, the k-th direct power of the simple linear group \(L_{n}(2)\) is uniquely determined by its spectrum in the class of finite groups provided n is a power of 2 greater...

Let G be a finite group and N(G) be the set of its conjugacy class sizes without 1. In the 1980s Thompson conjectured that the equality \(N(G)=N(S)\), where \(Z(G)=1\) and S is simple, implies the isomorphism \(G\simeq S\). In a series of papers of different authors, Thompson’s conjecture was proved. We show that if G is a finite centerless group w...

Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$, $N(B)=\{1,n\}$, and $n$ is a power of prime.

The question on connection between the structure of a finite group $G$ and the properties of the indices of elements of $G$ has been a popular research topic for many years. The $p$-index $|x^G|_p$ of an element $x$ of a group $G$ is the $p$-part of its index $|x^G|=|G:C_G(x)|$. The presented short note describes some new results and open problems...

The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer $k$, the $k$-th direct power of the simple linear group $L_{n}(2)$ is uniquely determined by its spectrum in the class of finite groups provided $n$ is a power of $2$ g...

Let $N(G)$ be a set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$ and $N(B)=\{1,n\}$, $n$ is a power of prime.

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that $Ind_G(a)_p\in \{1,p^{\alpha}\}$ for every $a$ of $G$ and a $p$-element $x\in G$ such that $Ind_G(x)_p>1$, then $G...

Axial algebras are commutative nonassociative algebras generated by a finite set of primitive idempotents which action on an algebra is semisimple, and the fusion laws on the products between eigenvectors for these idempotents are fulfilled. We find the sufficient conditions in terms of the Frobenius form and of the properties of idempotents under...

The spectrum of a finite group is the set of its element orders. We prove that if m>5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>5$$\end{document} then the group...

Let [Formula: see text] be a finite group and [Formula: see text] be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality [Formula: see text], where [Formula: see text] and [Formula: see text] is simple, implies the isomorphism [Formula: see text]. In a series of papers of different authors, Thompson’s conjectu...

We describe all degenerations of the variety [Formula: see text] of Jordan algebras of dimension three over [Formula: see text]. In particular, we describe all irreducible components in [Formula: see text]. For every [Formula: see text] we define an [Formula: see text]-dimensional rigid “marginal” Jordan algebra of level one. Moreover, we discuss m...

The spectrum of a finite group is the set of its element orders. In this paper we prove that the direct product of two copies of the finite simple sporadic group J4 is uniquely determined by its spectrum in the class of all finite groups.

A. Smoktunowicz and L. Vendramin conjectured that if A is a finite skew brace with solvable additive group, then the multiplicative group of A is solvable. In this short note we make a step towards positive solution of this conjecture proving that if A is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group...

In this paper we consider a prime graph of finite groups. In particular, we expect finite groups with prime graphs of maximal diameter.

The spectrum of a finite group is a set of its element orders. We prove that if $m>5$ then the group $L_{2^m}(2)\times L_{2^m}(2)\times L_{2^m}(2)$ is uniquely determined by its spectrum in the class of finite groups

Axial algebras of Jordan type η are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing (x−1)x(x−η), where η∉{0,1} is fixed, with restrictive multiplication rules. These properties generalize the Peirce decompositions for idempotents in Jordan algebras, where 12 is replaced with η. In particula...

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

A. Smoktunowicz and L. Vendramin conjectured that if $A$ is a finite skew brace with solvable additive group, then the multiplicative group of $A$ is solvable. In this short note we make a step towards positive solution of this conjecture proving that if $A$ is a minimal finite skew brace with solvable additive group and non-solvable multiplicative...

Axial algebras of Jordan type $\eta$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}...

We classify all complex 6-dimensional nilpotent Tortkara algebras.
Communicated by Alberto Facchini

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980's Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of different authors Thompson's conjecture was proved. In this paper, we show that in some cases it is possible t...

The greatest power of a prime p dividing the natural number n will be denoted by \(n_p\). For a set of primes \(\pi \) and a natural number n we will denote \(n_{\pi }=\prod _{p\in \pi }n_p\). Let G be a finite group with trivial center, and \(p,q>5\) be distinct prime divisors of |G|. We prove that if for every nonunity conjugacy classes size \(\a...

Let G be a finite group and N(G) be its set of conjugacy class sizes. We prove that if L is a finite simple non-abelian group and G is a finite group with trivial center such that N(G)=N(L) then G≃L.

We give a geometric classification of all 6-dimensional nilpotent Tortkara algebras over C.

A ring is said to be serial if its right and left regular modules are the direct sums of chain modules. The aim of the paper is to give an answer to the following question: for which finite simple groups, the group ring over a given field is serial.

We give a geometric classification of all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C

The spectrum of a finite group is the set of its element orders. In this paper we prove that the direct product of two copies of the finite simple sporadic group $J_4$ is uniquely determined by its spectrum in the class of all finite groups.

We classify all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C.

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that $Ind_G(a)_p\in \{1,p^{\alpha}\}$ for every $a$ of $G$ and a $p$-element $x\in G$ such that $Ind_G(x)_p>1$, then $G...

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that $Ind_G(a)_p\in \{1,p^{\alpha}\}$ for every $a$ of $G$ and a $p$-element $x\in G$ such that $Ind_G(x)_p>1$, then $G...

Given a finite group G, denote by N(G) the set of sizes of its conjugacy classes. We show that if G has trivial center and N(G) equals N(Altn) or N(Symn) for n≥23 then G has a composition factor isomorphic to the alternating group Altk for some k≤n such that the half-interval (k,n] contains no primes. As a corollary, we prove Thompson’s conjecture...

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group.

In this paper we describe all the finite almost simple groups whose Gruenberg--Kegel graphs coincide with Gruenberg--Kegel graphs of finite solvable groups.

Let $G\in\{p,q\}^*$ be a finite group with trivial center, where $p,q\in\pi(G)$ and $p>q>5$. In the present paper it is proved that $|G|_{\{p,q\}}=|G||_{\{p,q\}}$; in particular $C_G(g)\cap C_G(h)=1$ for every $p$-element $g$ and every $q$-element $h$.

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order $rs$. Let $A_n$ ($S_n$) denote the alternating (symmetric) group of degree $n$. We prove that if $G$ is a finit...

We describe all degenerations of the variety $\mathfrak{Jord}_3$ of Jordan algebras of dimension three over $\mathbb{C}.$ In particular, we describe all irreducible components in $\mathfrak{Jord}_3.$ For every $n$ we define an $n$-dimensional rigid Jordan algebra of level one.

Построена лупа Муфанг $M$ порядка $3^{19}$ и пара $a$, $b$ таких ее элементов, что множество всех элементов $M$, ассоциирующих с $a$ и $b$, не является подлупой. Это также дает пример неассоциативной лупы Муфанг с порождающим множеством, любые три элемента которого ассоциируют. Библиография: 5 названий.

A Moufang loop M of order 3¹⁹ is constructed, together with a pair a, b of elements of M, such that the set of all elements of M associating with a and b is not a subloop. This also gives an example of a nonassociative Moufang loop with a generating set in which every three elements have trivial associator.

Let $G$ be a finite group, and let $N(G)$ be the set of sizes of its conjugacy classes. We show that if a finite group $G$ has trivial center and $N(G)$ equals to $N(Alt_n)$ or $N(Sym_n)$ for $n\geq 23$, then $G$ has a composition factor isomorphic to an alternating group $Alt_k$ such that $k\leq n$ and the half-interval $(k, n]$ contains no primes...

The article is devoted to improving the lecture methods of the discipline «Methods of scientific research in tourism» for bachelors of the direction «Tourism». According to the requirements of Federal state educational standard the bachelor student must prepare for several types of activities that are in demand in the profession, including research...

For a finite group $G$, let $N(G)$ denote the set of conjugacy class sizes of $G$. We show that if every finite group $G$ with trivial center such that $N(G)$ equals to $N(Alt_n)$, where $n>1361$ and at least one of numbers $n$ or $n-1$ are decomposed into a sum of two primes, then $G\simeq Alt_n$.

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Altn) or N(Symn), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes.

The spectrum of a group is the set of its element orders. A finite group G is said to be recognizable by spectrum if every finite group with the same spectrum is isomorphic to G. We prove that if n ∈ P(15, 16, 18, 21, 27) then symmetric groups Symn are recognizable by spectrum.

We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its
elements such that the set of all elements of $M$ that associate with $a$ and
$b$ does not form a subloop. This is also an example of a nonassociative
Moufang loop with a generating set whose every three elements associate.

For a finite group $G$ denote by $N(G)$ the set of conjugesy class sizes of
$G$. We show that every finite group $G$ with the property $N(G)=N(Alt_n), n>4$
or $N(G)=N(Sym_n), n>22$ is non-solvable.

The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum if every finite group whose spectrum coincides with the spectrum of G is isomorphic to G. It is proved the symmetric group S n is recognizable by spectrum for n ∉ {2, 3, 4, 5, 6, 8, 10, 15, 16, 18, 21, 27, 33, 35, 39, 45}.

The spectrum of a group is the set of its element orders. A finite group $G$
is said to be recognizable by spectrum if every finite group that has the same
spectrum as $G$ is isomorphic to $G$. We prove that the simple alternating
groups $A_n$ are recognizable by spectrum when $n\neq 6, 10$. This implies that
every finite group with the same spectr...

We deal with finite simple groups G with the property π(G) ⊆ {2, 3, 5, 7, 11, 13, 17}, where π(G) is the set of all prime divisors of the order of a group G. The set of all such groups is denoted by ζ
17. Thompson’s conjecture in [1, Question 12.38] is proved valid for all groups in ζ
17 whose prime graph is connected.

The spectrum of a finite group is the set of its element orders. A group is said to be recognizable (by spectrum) if it is isomorphic to any finite group that has the same spectrum. A nonabelian simple group is called quasi-recognizable if every finite group with the same spectrum possesses a unique nonabelian composition factor and this factor is...

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum
of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple
groups by spectrum.

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.