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 G isospectral to L is squeezed between L and its automorphism group, that is L ≤ G ≤ Aut L; in particular, there are only finitely many such groups. This assertion with a series of previously obtained results yields that the same is true for every finite simple exceptional group except the group 3 D 4 (2).

On Recognizability of Groups by Bottom Layer

“…Let L be a finite simple exceptional group of Lie type, and L ≠ 3 D4 (2). Then any finite group isospectral to L is isomorphic to a finite group G, such that L ≤ G ≤ Aut L. In particular, L is almost recognizable by spectrum [22].…”

“…Proposition 1 (Vasiliev theorem [22]). Let G be a finite simple group U4 (5) and H a finite group with the property ω(H) = ω(G).…”

On the structure of finite groups isospectral to finite simple groups

“…The covers of other unitary and linear groups were settled in [30,39,127,130,133], of symplectic and orthogonal groups in [30,31]. The groups 3 D 4 (q) with q = 2, F 4 (q), E ε 6 (q) and E 7 (q) are recognizable among covers by [31,118]. Finally, 3 D 4 (2) is not recognizable among covers by [75].…”

Recognition of symplectic and orthogonal groups of small dimensions by spectrum

“…If G is a finite group having a nontrivial normal solvable subgroup, then by [1,Lemma 1], there are infinitely many pairwise nonisomorphic finite groups isospectral to G. In contrast, the finite nonabelian simple groups are rather satisfactorily determined by the spectrum. We refer to a nonabelian simple group L as recognizable by spectrum if every finite group G isospectral to L is isomorphic to L, and as almost recognizable by spectrum if every such a group G is an almost simple group with socle isomorphic to L. It is known that all sporadic and alternating groups, except for J 2 , A 6 and A 10 , are recognizable by spectrum (see [2,3]) and all exceptional groups excluding 3 D 4 (2) are almost recognizable by spectrum (see [4,5]). In 2007 V.D.…”