The group J4 × J4 is recognizable by spectrum

Abstract

The spectrum of a finite group is the set of its element orders. In this paper we prove that the direct product of two copies of the finite simple sporadic group J 4 is uniquely determined by its spectrum in the class of all finite groups.In proving Theorem, we use the following assertion which is interesting in its own right.

Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

“…Note that a group with non-simple socle can be recognizable by spectrum, therefore Theorem 1.3 can not be generalized for recognition by spectrum. Up to a recent moment there were only two examples of groups with non-simple socle which are recognizable by spectrum, namely, Sz(2 7 ) × Sz(2 7 ) (see [66]) and J 4 × J 4 (see [24]). Recently, I.…”

Criterion of nonsolvability of a finite group and recognition of direct squares of simple groups

“…Despite the fact that there are a huge number of simple groups recognizable by spectrum (see [1,Theorem 1]), there are only two examples of a recognizable group that is a direct square of a simple group: Sz(2 7 ) × Sz(2 7 ) [6] and J 4 × J 4 [4]. The reason for this is that the standard methods of proving the nonsolvability of groups isospectral to a simple group are based on properties of its prime graph.…”

“…Clearly, they are not applicable to proving the nonsolvability of a group isospectral to the square of a simple group, since in this case the corresponding graph is always complete. Proposition 1 of [4], mentioned in Remark 3, is a significant advance in this direction, but it is difficult to verify the conditions of this proposition for squares of arbitrary simple groups. As Corollary 2 shows, Theorem 1 of the present paper does not have this disadvantage and therefore allows us to hope for new examples of recognizable groups that are the squares of simple groups.…”

Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph

“…Note that a group with non-simple socle can be recognizable by spectrum, therefore Theorem 1.3 can not be generalized for recognition by spectrum. Up to a recent moment there were only two examples of groups with non-simple socle which are recognizable by spectrum, namely, Sz(2 7 ) × Sz(2 7 ) (see [32]) and J 4 × J 4 (see [8]). Recently, I.…”