On finite rational groups and related topics. (English) Zbl 0701.20005

In this interesting paper, the authors use the classification of finite simple groups to settle a number of questions about finite groups that have been outstanding for many years. Given a finite group \(H\), let \({\mathbb{Q}}(H)\) denote the field generated over the rational field \({\mathbb{Q}}\) by the values of all the complex irreducible characters of \(H\). \(H\) is said to be a rational group if \({\mathbb{Q}}(H)={\mathbb{Q}}\). The authors show that, given a positive integer \(m\), there is a finite set \(F_ m\) of simple groups such that if \(G\) is a finite noncyclic simple groups that occurs as a composition factor of a group \(H\) satisfying [\({\mathbb{Q}}(H):{\mathbb{Q}}]\leq m\), then \(G\) is isomorphic to an alternating group or to a group in \(F_ m\). There remains open the question of whether there are only finitely many cyclic groups of prime order that can occur as composition factors of a group \(H\) satisfying the condition above. The authors also determine explicitly which noncyclic finite simple groups can occur as composition factors of rational groups. These are: the alternating groups, \(\text{PSp}(4,3)\), \(\text{Sp}(6,2)\), \(\text{SO}^+(8,2)\), \(\text{PSL}(3,4)\), \(\text{PSU}(4,3)\). All but the last two groups in this list occur as composition factors of Weyl groups and, moreover, it is shown in the paper that \(\text{Sp}(6,2)\) and \(\text{SO}^+(8,2)\) are the only noncyclic simple groups that are rational groups. That a certain extension of \(\text{PSL}(3,4)\) by an automorphism of order 2 is a rational group can be seen in the Atlas of finite groups [J. H. Conway et al., Oxford (1985; Zbl 0568.20001)]. The extension of \(\text{PSU}(4,3)\) by an elementary Abelian 2-group of order 4 is a more surprising rational group. We might call a rational group strongly rational if it has \({\mathbb{Q}}\) as a splitting field. The Weyl groups are strongly rational, as is the extension of \(\text{PSL}(3,4)\). It would be interesting to know whether the rational group derived from \(\text{PSU}(4,3)\) is strongly rational. The reviewer proved that the only cyclic groups of prime order that occur as composition factors of strongly rational groups are those of order 2 and 3.
The authors also solve a well-known problem on finite groups that had been investigated by G. A. Miller in 1933. Specifically, if \(G\) is a finite group in which any two elements of the same order are conjugate, then \(G\) is isomorphic to one of the symmetric groups of degree 1, 2, or 3. A previous proof is due to P. Fitzpatrick [Proc. R. Ir. Acad., Sect. A 85, 53-58 (1985; Zbl 0558.20016)]. In addition, a question of G. Janusz is answered by showing that the rational group algebra \({\mathbb{Q}}[G]\) of any nontrivial finite group \(G\) has an outer automorphism. We remark that many of the technicalities of the proof involve proving the existence of cyclic self-centralizing maximal tori in finite simple groups of Lie type. These subgroups generalize the well-known Singer cycles in classical groups.


20C15 Ordinary representations and characters
20C33 Representations of finite groups of Lie type
20D05 Finite simple groups and their classification
20C05 Group rings of finite groups and their modules (group-theoretic aspects)