Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, September–October, 1997.     

Recognition of finite groups by the prime graph

We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 390–408, July–August, 2006.     

A characterization of finite nonsimple groups by the set of orders of their elements

It is proved that the permutation wreath product H of a simple Suzuki group Sz(2⁷) and a subgroup fo a symmetric group of degree 23, isomorphic to a Frobenius group of order 253, is (up to isomorphism) distinguished among all finite groups by the set of orders of its elements. Since H possesses a minimal normal subgroup N that contains an element of order equal to the exponent of N, this result furnishes a counterexample to one of the conjectures set forth by Shi [1]. In addition, we show that the direct square of a group Sz(2⁷) is also distinguished by the set of orders of its elements. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 304–322, May–June, 1997.     

Generation of finite groups by the class of conjugate Abelian subgroups

Using the basic theorem on the classification of finite simple groups, we answer one of the questions concerning the generation of finite groups by the class of conjugate Abelian subgroups. Supported by RFFR grant No. 93-01-01529. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 288–293, May–June, 1996.     

Infinite groups with Abelian centralizers of involutions

The article contains two characterizations of projective linear groups PGL₂(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second, in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which are recognizable by the set of their element orders. In memory of Viktor A. Gorbunov Supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 74–86, January–February, 2000.     

Minimal permutation representations of finite simple orthogonal groups

In the paper, nontrivial permutation representations of minimal degree are studied for finite simple orthogonal groups. For them, we find degrees, ranks, subdegrees, point stabilizers and their pairwise intersections.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 603–627, November–December, 1994.     

Recognition of simple groups U 3(Q) by element orders

An exhaustive solution is given to the recognition-by-spectrum problem for finite, simple, three-dimensional unitary groups. For every such group, the number of non-isomorphic, finite, isospectral groups is determined. In particular, a new counterexample to Problem 13.63 in the Kourovka Notebook is furnished. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 185–202, March–April, 2006.     

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|²<|G|. Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000.     

The cyclic structure of maximal tori of the finite classical groups

Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form. Supported by RFBR grant Nos. 05-01-00797 and 06-01-39001, and by SB RAS Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 129–156, March–April, 2007.     

Carter subgroups of finite almost simple groups

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.     

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.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.     

Locally nilpotent groups with the minimal condition on centralizers

It is proved that locally nilpotent groups with the minimal condition on centralizers are hypercentral, and that the Fitting subgroup of a group with the minimal condition on centralizers of normal subgroups is nilpotent. Supported by RFFR grant No. 96-01-00358. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 270–278, May–June, 1998.     

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem. Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No. 2006.1.2). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008.     

Associative nil-algebras and Golod groups

Non-nilpotent, finitely generated, associative nil-algebras are studied as well as their adjoint groups and Golod groups. Solutions are given to some problems in residually finite group theory, questions posed in the Kourovka Notebook included. Supported by RFBR grant No. 03-01-00356. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 231–238, March–April, 2006.     

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.