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**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.

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.

### MSC:

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) |