2-FROBENIUS Q-GROUPS

摘要

Let G be a finite group. G is called a rational group or a Q-group if every complex irreduciblecharacter of G is rational valued. Equivalently G is a Q-group if and only if every x in G isconjugate to x "2, where m E N and (o(x), m) = 1. This means that for every x E G of ordern, we have Nc((x))/CG((x)) '1== Aut((x)), where Aut((x)) is a group of order (p(n) where coois the Euler function. The symmetric group Sri and the Weyl groups of complex Lie algebras areexamples of Q-groups 2. Q-groups have been studied extensively; but classifying finite Q-groupsis still an open research problem. It has been shown in 5 that if G is a solvable Q-group then7r(G) C {2, 3,5}, where 7r(G) denotes the set of prime divisors of 'GI. The structure of FrobeniusQ-groups have been described in 3. It is proved in 7 that if G is a solvable Q-group then itsSylow 5-subgroup is normal and elementary abelian.