Let p > 2 and l be distinct primes, let G be a finite l-solvable group and let P ? Syl(p)(G). Then G has a unique irreducible l-Brauer character of p'-degree lying over 1(P) if and only if N-G(P)/P is an l-group. This extends a result from [J. F. Tent, Correspondences of Brauer characters and Sylow subgroup normalizers, J. Algebra 573 (2021) 436-450] to every odd prime p.

Let [Formula: see text] and [Formula: see text] be distinct primes, let [Formula: see text] be a finite [Formula: see text]-solvable group and let [Formula: see text]. Then [Formula: see text] has a unique irreducible [Formula: see text]-Brauer character of [Formula: see text]-degree lying over [Formula: see text] if and only if NG (P)/ P is an [Formula: see text]-group. This extends a result from [J., F.,Tent, Correspondences of Brauer characters and Sylow subgroup normalizers, J. Algebra 573 (2021) 436–450] to every odd prime [Formula: see text].

Let p>3 and q≠p be primes, let G be a finite q-solvable group and let P∈Sylp(G). Then G has a unique irreducible q-Brauer character of p′-degree lying over 1P if and only if NG(P)/P is a q-group. One direction of this result follows from a natural McKay bijection of p′-degree irreducible q-Brauer characters, which is obtained under suitable conditions.

We show that every solvable group is a subgroup of some monomial real group. This extends a result of Dade, who proved that every solvable group is a subgroup of a monomial group.

Let $G$ be a finite group, $p$ a prime and $P$ a Sylow $p$-subgroup of $G$. In this note we give a cohomological criterion for the $p$-solvability of $G$ depending on the cohomology in degree $1$ with coefficients in $\mathbb F_p$ of both the normal subgroups of $G$ and $P$. As a byproduct we bound the minimal number of quotients of order a power of $p$ appearing in any normal series of $G$ by the number of generators of $P$.

Let G be a finite solvable or symmetric group, and let B be a 2-block of G. We construct a canonical correspondence between the irreducible characters of height zero in B and those in its Brauer first main correspondent. For symmetric groups our bijection is compatible with restriction of characters.

Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of NG(P ), where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if k is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly k real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-p techniques, and, in particular, we use the Kneser classification of semi-simple p-adic algebraic groups.

Let G be a p-solvable group of p-length l, where p is any prime. We show that G has at least 2(l) irreducible characters of degree coprime to p and having values inside Q(p). This generalizes a previous result for p=2 [6] to arbitrary primes. With the same notation, we prove that if p is odd then G has at least 2(l) Galois orbits of conjugacy classes of p-elements having values in Q(p).

We study the finite 2-groups with a fixed number of real conjugacy classes. The order of such groups can be arbitrarily large but we show that it can be bounded if the orders of the elements in a generating set are also fixed. If the number k of real classes is odd we show that the group order can be bounded in terms of k and the nilpotency class although we conjecture that a bound in terms only of k exists. We confirm this conjecture when k=7.

We give the lower bounds 2ℓ and 2ℓ−2+2ℓ−4+⋯ for the number of irreducible characters of height zero and positive height, respectively, in a p-block of a p-solvable group, where ℓ is the p-length of an associated p-solvable group. We also prove some results on extensions of linear characters in p-subgroups of p-solvable groups.

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.

If G is a finite solvable group of 2-length l, we prove that the number of odd degree rational-valued irreducible characters of G is at least 2 l , improving a result of G. Navarro and P. H. Tiep. This bound is best possible, and also provides a new global/local relationship.

In this paper we prove the following conjecture of G. Navarro: if G is a finite 2-group with exactly 5 rational conjugacy classes, then G is dihedral, semidihedral or generalized quaternion. We also characterize the 2-groups with 4 rational classes.

Let G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.