On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

Parabolic permutation representations of the groups2 F 4(q) and3 D 4(q 3)

In the paper, the ranks, degrees, subdegrees, and double centralizers of permutation representations of the bounded groups² F ₄(q) and³ D ₄(q ³) with respect to parabolic maximal subgroups of nonminimal index are found. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 69–76, January, 2000.     

On recognition of finite simple groups with connected prime graph

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.     

On Connection Between the Structure of a Finite Group and the Properties of Its Prime Graph

It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg-Kegel graph of a finite group G under some natural additional conditions suffices to describe the structure of G; in particular, to prove that G has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.Original Russian Text Copyright © 2005 Vasilev A. V.The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8294), the Program Universities of Russia (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 511–522, May–June, 2005.     

The Cayley graphs of ℤ<Superscript>d</Superscript> and the limits of vertex-primitive graphs of <Emphasis Type="Italic">HA</Emphasis>-type

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups ?^d. In this article we prove that for each d > 1 the set of Cayley graphs of ?^d presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for d < 4 we list all Cayley graphs of ?^d that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ?^d with crystallographic groups.     

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.     

Recognizability of finite simple groups <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) and <Emphasis Type="Italic">U</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) by spectrum

It is proved that finite simple groups L₄(2^m), m ⩾ 2, and U₄(2^m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8. Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry of Education of China (Project for Retaining Foreign Expert). Supported by NSF of Chongqing (CSTC: 2005BB8096). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

关于216阶群的完全分类

设G为2~3·3~3阶(即216阶)群,本文研究G的同构分类.利用有限群的局部分析法,证明G共有177种互不同构的类型,并获得了G的全部构造.     

On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

On the Dimension of Nilpotent Algebras

The Eggert conjecture claims that a finite commutative algebra R over a field of prime characteristic p has the property dim Rdim R⁽¹), where R(1) is the subspace of R spanned by the pth powers of elements of R. We obtain results related to this conjecture and results on nilpotent algebras of rather high nilpotency class.     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

OD-Characterization of Some Projective Special Linear Groups Over the Binary Field and Their Automorphism Groups

The Gruenberg–Kegel graph GK(G) = (V _G , E _G ) of a finite group G is a simple graph with vertex set V _G  = π(G), the set of all primes dividing the order of G, and such that two distinct vertices p and q are joined by an edge, {p, q} ∈ E _G , if G contains an element of order pq. The degree deg _G (p) of a vertex p ∈ V _G is the number of edges incident to p. In the case when π(G) = {p ₁, p ₂,…, p _h } with p ₁ < p ₂ < … <p _h , we consider the h-tuple D(G) = (deg _G (p ₁), deg _G (p ₂),…, deg _G (p _h )), which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying condition (|H|, D(H)) = (|G|, D(G)). Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we prove that the simple groups L ₁₀(2) and L ₁₁(2) are OD-characterizable. It is also shown that automorphism groups Aut(L _p (2)) and Aut(L _p+1(2)), where 2 ^p  ? 1 is a Mersenne prime, are OD-characterizable. Finally, a list of finite (simple) groups which are presently known to be k-fold OD-characterizable, for certain values of k, is presented.     

On Hall Normally Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H ^G . Groups which contain a Hall normally embedded subgroup of order d for every factor d of | G | are characterized. Such groups are supersolvable with a cyclic nilpotent residual of square-free order.     

Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

We consider a finite matrix group with 3⁴· 2¹⁶ elements, which is a subgroup of the infinite group , where is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically. We study the structure of the group and use this group to construct spherical orbit codes on the unit Euclidean sphere in R⁸. These codes have code distance less than 1. One of them has 3²· 2⁸ = 2304 elements and its squared Euclidean code distance is 0.293. Communicated by: V. A. Zinoviev     

Recognition of the Group O+ 10(2) from Its Spectrum

We prove that if the set of orders of elements of a finite group G coincides with the set of orders of elements of the group D=O₁₀ ⁺(2), then G is isomorphic to D. In other words, O ₁₀ ⁺(2) is recognizable from its spectrum.