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.     

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.     

Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.     

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26. Supported by RFBR (project Nos. 08-01-00322 and 06-01-39001), by SB RAS (Integration Project No. 2006.1.2), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and Young Doctors and Candidates of Science (grants MD-2848.2007.1 and MK-377.2008.1). Translated from Algebra i Logika, Vol. 47, No. 5, pp. 558–570, September–October, 2008.     

On periodic groups with small orders of elements

We prove that a finite group with the set of element orders equal to {1, 2, 3, 5, 6} is locally finite.     

Pure quantitative characterization of finite simple groups

Let G be a finite group, and let π _e (G) be the set of all element orders of G. In this short paper we prove that π _e (B _n (q)) ≠ π _e (C _n (q)) for all odd q.      

A characterization of the simple group PSL<Subscript>5</Subscript>(5) by the set of its element orders

Let G be a finite group and let ω(G) denote the set of the element orders of G. For the simple group PSL₅(5) we prove that if G is a finite group with ω(G) = ω(PSL₅(5)), then either G ? PSL₅(5) or G ? PSL₅(5): 〈θ〉 where θ is a graph automorphism of PSL₅(5) of order 2.     

Pure quantitative characterization of linear groups over the binary field

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut (L ₅(2)) and the linear group L ₆(2). It is proved that Aut(L ₅(2)) and L ₆(2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William’s work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups. __________ Translated from Chinese Annals of Mathematics, 2003, 24A(6): 675–682. This work was supported by the National Natural Science Foundation of China under grant number 10171074     

The <Emphasis Type="Italic">F</Emphasis>-local subgroups of the groups with a generalized finite element of order 2 or 4

We prove the existence of infinite subgroups with nontrivial locally finite radicals and of infinite locally finite subgroups in the groups with almost finite almost solvable elements of order 2 and 4 and in the groups with almost finite elements.     

Second maximum sum of element orders on finite groups

Amiri, Jafarian Amiri and Isaacs proved that the cyclic group has maximum sum of element orders on all groups of the same order. In this article we characterize finite groups which have maximum sum of element orders among all noncyclic groups of the same order. This result confirms the conjecture posed in Jafarian Amiri (2013) [2].     

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.     

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 groups

It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained.     

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.     

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.     

Characterization of the finite variation property for a class of stationary increment infinitely divisible processes

We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Lévy processes, and also their mixtures. We establish two types of zero–one laws for the finite variation property. We also consider some examples to illustrate our results.     

Large element orders and the characteristic of finite simple symplectic groups

The characteristic of a simple group of Lie type is the characteristic of the field over which this group is defined. Let G = Sp_2n (q), where q = 2 ^k . It is shown that every finite group of Lie type with the same two largest element orders as G has characteristic 2.