Characterization of the finite simple groups by spectrum and order

Abstract

We give an affirmative answer to Question 12.39 in the Kourovka Notebook. Namely, it is proved that a finite simple group and a finite group having equal orders and same sets of element orders are isomorphic.The spectrum of a group G is the set ω(G) of its element orders. The spectrum of a finite group G together with its order retains a substantial part of information on the structure of G but, as demonstrated by the example of the dihedral group D 8 of order 8 and the quaternion group Q 8 , does not necessarily determine G uniquely. In [1], W. Shi conjectured that the desired uniqueness would be achieved for finite simple groups. In [2, Question 12.39], A. S. Kondratiev worded this conjecture as follows:Is it true that a finite group and a finite simple group are isomorphic if they have equal orders and same sets of element orders?Later, for brevity, it was suggested to refer to a finite group G that is isomorphic to every finite group H with ω(H) = ω(G) and |H| = |G| as recognizable by spectrum and order. In this terminology, Shi's question reads as follows:Is it true that all finite simple groups are recognizable by spectrum and order?The answer to this question is obviously 'yes' for Abelian simple groups. In a series of papers [1,[3][4][5][6][7][8], an affirmative answer was given for all non-Abelian simple groups except the symplectic groups, orthogonal groups of odd dimension, and orthogonal groups of type D n with n even. In the present paper, we argue for the recognizability by spectrum and order for these remaining groups. Thus, Shi's conjecture is confirmed and the following theorem holds true.

A CHARACTERIZATION OF SOME PGL(2, q) BY MAXIMUM ELEMENT ORDERS

“…For example, it is proved recently by Shi and Mazurov et al that every finite simple group S can be determined by |S| and π e (S) (see [2,10,18,19,21,22,23,24,25,26,28]). This means that every simple group can be characterized by pure quantitative properties.…”

Order classes of dihedral groups

“…The number of distinct rotations is n which is also the number of distinct reflections, so n D n 2 2 . In general, let 1 Throughout all, we follow this representation for dihedral groups.…”

“…In [1] , it was proven that a finite simple group and a finite group having equal orders and the same sets of element orders are isomorphic.…”

“…All finite simple groups may be determined by their group order and the sets of element orders [1]. Moreover, it was proven that a finite simple group and a finite group having equal orders and same sets of element orders are isomorphic.…”

Finite groups having the same prime graph as the group Aut(J 2)

“…The proof of the recognizability of any finite simple group by its order and spectrum has been completed recently (see [1]). The conjecture that finite simple groups are, as a rule, recognizable by their order and prime graph seems to be quite plausible.…”