On finite simple groups with the set of element orders as in a Frobenius group or a double Frobenius group. (English. Russian original) Zbl 1065.20025

Aleeva, M. R.

Math. Notes 73, No. 3, 299-313 (2003); translation from Mat. Zametki 73, No. 3, 323-339 (2003).

Using the classification, it is proved that the only finite simple groups having the same element orders as a Frobenius group are \(L_3(3)\) and \(U_3(3)\). Examples of corresponding Frobenius groups are specific groups of shape \(13^2:2S^-_4\) and \(7^4:3:8\), respectively. Similarly, the only possible finite simple groups having the same element orders as a double Frobenius group (a group of shape \(A:B:C\) where \(A:B\) and \(B:C\) are Frobenius groups) are \(U_3(3)\) and \(S_4(3)\). No double Frobenius groups with these sets of element orders are apparently known.

Reviewer: Robert Wilson (London)

MSC:

20D06	Simple groups: alternating groups and groups of Lie type
20D60	Arithmetic and combinatorial problems involving abstract finite groups

Keywords:

finite simple groups; double Frobenius groups; sets of element orders