Alexey Staroletov

• PhD
• Sobolev Institute of Mathematics

Denote the symmetric group of degree $n$ by $S_n$. If $\sigma\in S_n$ and $\rho$ is an irreducible representation of $S_n$ over the field of complex numbers, then we describe the set of eigenvalues of $\rho(\sigma)$.

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...

The Gruenberg–Kegel graph (or the 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 order rs in G. A finite group G is called almost recognizable (by Gruenberg–Kegel graph) if there is onl...

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...

The Gruenberg--Kegel 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 order $rs$ in $G$. A finite group $G$ is called {\it almost recognizable} (by Gruenberg--Kegel graph) ifthere is only f...

The Gruenberg-Kegel 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 order $rs$ in $G$. A finite group $G$ is called almost recognizable (by Gruenberg-Kegel graph) if there is only finite n...

Axial algebras are a class of commutative algebras generated by idempotents, with adjoint action semisimple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren, and Shpectorov in 2015 as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for t...

Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement for the torus $T$ in its algebraic normalizer $N(G,T)$. Namely, it is proved that any maximal torus of the group $G\in\{G_2(q), ^2G_2(q), ^3D_4(q), ^2F_4(q), ^2B_2(q)\}$ has a complement in its a...

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 be a finite group of Lie type and the Weyl group of . For every maximal torus of , we find the minimal order of a supplement of in its algebraic normalizer . In particular, we find all the maximal tori that have a complement in . Let correspond to an element of . We find the minimal orders of the lifts of the elements in .

Пусть $G$ - конечная группа лиева типа $F_4$ и $W$ - группа Вейля группы $G$. Для каждого максимального тора $T$ группы $G$ найден минимальный порядок добавления к тору $T$ в его алгебраическом нормализаторе $N(G,T)$. В частности, найдены все максимальные торы, имеющие дополнение в группе $N(G,T)$. Пусть тор $T$ соответствует элементу $w$ группы $W...

The Gruenberg-Kegel graph (or the prime graph) Γ(G) of a finite group G is dened as follows. The vertex set of Γ(G) is the set of all prime divisors of the order of G. Two distinct primes r and s regarded as vertices are adjacent in Γ(G) if and only if there exists an element of order rs in G. Suppose that L \cong E6(3) or L \cong 2E6(3). We prove...

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as vertices are adjacent in $\Gamma(G)$ if and only if there exists an element of order $rs$ in $G$. Suppose that $L\con...

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the propert...

Isospectral are the groups with coinciding sets of element orders. We prove that no finite group isospectral to a finite simple classical group has the exceptional groups of types \( E_{7} \) and \( E_{8} \) among its nonabelian composition factors.

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...

Denote the alternating and symmetric groups of degree n by An and Sn, respectively. Consider a permutation σ ∈ Sn, all of whose nontrivial cycles are of the same length. We find the minimal polynomials of σ in the ordinary irreducible representations of An and Sn.

Let $G$ be a finite group of Lie type $F_4$ with the Weyl group $W$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement to $T$ in its algebraic normalizer $N(G,T)$. In particular, we obtain all maximal tori having complements in $N(G,T)$. Assume that $T$ corresponds to an element $w$ of $W$. We find the minimal order of l...

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}...

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the p...

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$, respectively. Consider a permutation $\sigma\in S_n$ whose all nontrivial cycles have the same length. We find minimal polynomials of $\sigma$ in ordinary irreducible representations of $A_n$ and $S_n$.

Let G be a finite group of Lie type E6 over q (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w W such that N(G, T)/T CW(w). When T does not have a complement isomorphic...

Majorana theory was introduced by A.A. Ivanov as the axiomatization of certain properties of the 2A-axes of the Griess algebra. Since its inception, Majorana theory has proved to be a remarkable tool with which to study objects related to the Griess algebra and the Monster simple group. We introduce the definition of a minimal 3-generated Majorana...

Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$ and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. When $T$ does not have a complement, we show that $w$ has a lift in $N(G,T)$ of order...

Majorana theory was introduced by A. A. Ivanov as the axiomatization of certain properties of the 2A-axes of the Griess algebra. Since its inception, Majorana theory has proved to be a remarkable tool with which to study objects related to the Griess algebra and the Monster simple group. We introduce the definition of a minimal 3-generated Majorana...

Let $G$ be a finite group of Lie type $E_6$ over $F_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that for each maximal torus $T$ of $G$ there exists an element $w\in W$ such that $N(G,T)/T\simeq C_W(w)$. When $T$...

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...

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...

The set of element orders of a finite group G is called the spectrum. Groups with coinciding spectra are said to be isospectral. It is known that if G has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic groups isospectral to G. The situation is quite different if G is a nonabelain simple group. Recently...

The spectrum of a finite group is the set of its elements orders. A group G is said to be unrecognizable by spectrum if there are infinitely many pairwise non-isomorphic finite groups having the same spectrum as G. We prove that the simple orthogonal group O_9(q) has the same spectrum as VO_8^-(q), where V is the natural 8-dimensional module of the...

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group $G$ isospectral to $L$ is squeezed between $L$ and its automorphism group, that is $L\leq G\leq \operatorname{Aut}L$; in particular, ther...

Two groups are said to be isospectral if they have equal sets of element orders. It is proved that for every finite simple exceptional group L = G 2(q) of Lie type, any finite group G isospectral to L must be isomorphic to L.

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B 3(q), C 3(q), and D 4(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed...

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U 4(2), U 5(2) then this factor is isomorphic to either L or a group of Lie ty...

This paper considers a group which has a nonabelian sporadic composition factor and the same set of element orders as a finite simple group. It is proved that such group is isomorphic to U 5(2) or a sporadic group.

The spectrum of a finite group is the set of its element orders. We describe the composition structure of every finite group with the same spectrum as that of the alternating group of degree 10 and not isomorphic to it. This group is isomorphic to the semidirect product of the abelian {3, 7}-group, which contains an element of order 21, by the symm...

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...