Composition Factors of the Finite Groups Isospectral to Simple Classical Groups

Siberian Mathematical Journal - 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...     

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 ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

Groups isospectral to the degree 10 alternating 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 symmetric group of degree 5.     

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

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 from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.