A. V. Vasil’ev

On finite groups isospectral to simple linear and unitary groups

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 ≠ U4(2), U5(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.

Recognition by spectrum for simple classical groups in characteristic 2

A finite group G is said to be recognizable by spectrum if every finite group with the same set of element orders as G is isomorphic to G. We prove that all finite simple symplectic and orthogonal groups over fields of characteristic 2, except S4(q), S6(2), O8+ (2), and S8(q), are recognizable by spectrum. This result completes the study of the recognition-by-spectrum problem for finite simple classical groups in characteristic 2.

Almost Recognizability by Spectrum of Simple Exceptional Groups of Lie Type

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 = E7(q), we prove that each finite group isospectral to L is isomorphic to a group G squeezed between L and its automorphism group, i.e., L ≤ G ≤ AutL; in particular, up to isomorphism, there are only finitely many such groups. This assertion, together with a series of previously obtained results, implies that the same is true for every finite simple exceptional group except the group3D4(2).

Locally Finite Groups with Bounded Centralizer Chains

The c-dimension of a group G is the maximal length of a chain of nested centralizers in G. We prove that a locally finite group of finite c-dimension k has less than 5k non-Abelian composition factors.

Testing Isomorphism of Central Cayley Graphs Over Almost Simple Groups in Polynomial Time

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly (n).