In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of the Galois group on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

pdf/galois-automorphisms-and-classical-groups-33lc3doccb.pdf

Galois Automorphisms and Classical Groups

Abstract Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G / P {G/P} is nilpotent if and only if φ ⁢ ( 1 ) 2 {\\varphi(1)^{2}} divides | G : ker ( φ ) | {|G:{\\rm ker}(\\varphi)|} for every irreducible Brauer character φ of G.

Blocks of small defect in alternating groups and squares of Brauer character degrees

We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where , we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.

/pdf/the-fields-of-values-of-characters-of-degree-not-divisible-4cr7u7vh15.pdf

The fields of values of characters of degree not divisible by p

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups. 

/pdf/galois-automorphisms-and-classical-groups-125f6st6.pdf

Galois automorphisms and classical groups

Nonabelian fully-ramified sections

Characters of Solvable Groups

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Characters of Solvable Groups