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RESEARCH PRODUCT

Quadratic rational solvable groups

Joan Tent

subject

Discrete mathematicsFinite groupAlgebra and Number TheoryField (mathematics)Isotropic quadratic formPrime (order theory)CombinatoricsQuadratic equationSolvable groupSolvable groupRational characterBounded functionQuadratic fieldQuadratic fieldMathematics

description

Abstract A finite group G is quadratic rational if all its irreducible characters are either rational or quadratic. If G is a quadratic rational solvable group, we show that the prime divisors of | G | lie in { 2 , 3 , 5 , 7 , 13 } , and no prime can be removed from this list. More generally, if G is solvable and the field Q ( χ ) generated by the values of χ over Q satisfies | Q ( χ ) : Q | ⩽ k , for all χ ∈ Irr ( G ) , then the set of prime divisors of | G | is bounded in terms of k . Also, we prove that the degree of the field generated by the values of all characters of a semi-rational solvable group (see Chillag and Dolfi, 2010 [1] ) or a quadratic rational solvable group over Q is bounded, giving a positive answer to a question by D. Chillag and S. Dolfi.

yearjournalcountryeditionlanguage
2012-08-01Journal of Algebra
10.1016/j.jalgebra.2012.04.019http://dx.doi.org/10.1016/j.jalgebra.2012.04.019