Geodesic
Almost recognizability by spectrum of simple exceptional groups of Lie type
Algebra i logika, Tome 53 (2014) no. 6, pp. 669-692

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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, i.e., $L\le G\le\mathrm{Aut}\,L$; 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 $^3D_4(2)$.
Keywords: finite simple groups, exceptional groups of Lie type, element orders, prime graph, recognition by spectrum. 
@article{AL_2014_53_6_a0,
     author = {A. V. Vasil'ev and A. M. Staroletov},
     title = {Almost recognizability by spectrum of simple exceptional groups of {Lie} type},
     journal = {Algebra i logika},
     pages = {669--692},
     publisher = {mathdoc},
     volume = {53},
     number = {6},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2014_53_6_a0/}
}
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A. V. Vasil'ev; A. M. Staroletov. Almost recognizability by spectrum of simple exceptional groups of Lie type. Algebra i logika, Tome 53 (2014) no. 6, pp. 669-692. http://geodesic.mathdoc.fr/item/AL_2014_53_6_a0/