Almost recognizability by spectrum of simple exceptional groups of Lie type

Abstract

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 $L=E_7(q)$, we prove that each finite group isospectral to $L$ is isomorphic to a group $G$ squeezed between $L$ and its automorphism group, that is $L\le G\le \mathrm{Aut}L$; 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 group ${}^3{D}_4(2)$.

minor changes, Tables 2 and 3 are fixed

References in corpus (1)

• On element orders in covers of finite simple groups of Lie type

Cited by in corpus (3)

• Finite groups isospectral to simple groups
• On the nilpotency of the solvable radical of a finite group isospectral to a simple group
• On recognition of symplectic and orthogonal groups of small dimensions by spectrum