We determine the fields of values of the Isaacs' head characters of a finite solvable group.

Very recently, the first and third authors proposed a new conjecture on characters of finite groups, related to the McKay conjecture. Let p be a prime number and G a finite group. We say that a p-element x ∈ G is picky in G if it is contained in a unique Sylow p-subgroup of G. The simplest formulation of this conjecture predicts the existence of a bijection between the set of irreducible characters of G that do not vanish on a picky p-element x, and the corresponding set of irreducible characters of the normalizer of the unique Sylow p-subgroup of G containing x. Moreover, this bijection is expected to satisfy several natural conditions. For example, the p-parts of the degrees of corresponding characters should coincide, and their values at x should also be suitably related. In this paper, we prove this conjecture for p-solvable groups when p is odd.

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ alternative group-theoretical techniques to address the "going down" case. This approach enables us to reduce the block version of well-known results by the third and fourth authors to a question about almost simple groups. Moreover, this suggests a Galois analogue of the height-zero-equal-degree conjecture of Malle and Navarro, which we formulate. However, the "going up" case of irreducible extensions of principal block characters remains unresolved.

We propose that characters induced from cyclic subgroups of a finite group G contain special irreducible constituents, whose existence would follow from a new property of characters of quasi-simple groups. This result would, in particular, imply Qian's main conjecture in [24], thereby reducing it to a question about quasi-simple groups. We also explore generalizations in which the inducing subgroup is nilpotent rather than cyclic, and induction is replaced by restriction. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We study the sum of the squares of the irreducible character degrees not divisible by some prime p, and its relationship with the the corresponding quantity in a p-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for p=2 and in some other cases.

Let G be a finite group and let p be a prime. If P is a nonabelian Sylow p-subgroup of G and m(P) is the smallest non-linear irreducible character degree of P, we prove that there exists χ∈ Irr(G) in the principal p-block of G such that 1<χ(1)_p≤ m(P), giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked “prime by prime". For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes p>5 we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.

Let p be a prime. For p=2, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by p have been classified; for odd primes p, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of p, and we provide some evidence supporting its validity.

Let P_n be a Sylow p-subgroup of the symmetric group S_n. We investigate the number and sizes of the P_n∖ S_n / P_n double cosets, showing that most double cosets have maximal size when p is odd, or equivalently, that P_n∩ P_n^x=1 for most x∈ S_n when n is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

Let $χ$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $χ(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of order 69120 such that this proportion is 511/1152.

The Gluck-Wolf theorem and its general version [Navarro and Tiep, Annals of Math. 178 (2013), 1135-1171] relate arithmetic properties at a fixed prime p of the ratios x(1)/lambda(1), for irreducible characters X of a finite group G that lie over a fixed -invariant irreducible character lambda of a normal subgroup Z of G, to the structure of Sylow p-subgroups of G/Z. This result constituted a key step towards the recent proof [Malle et al., Annals of Math. 200 (2024), 557-608] of Brauer's Height Zero Conjecture. In this paper, we prove a further extension of the Gluck-Wolf theorem to sets pi of primes, with a mild condition on pi if the alternating group A(7) is involved in the group.

We prove that the Eaton-Moret & oacute; Conjecture is true for the principal blocks of the p-solvable groups. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We prove that if all the simple groups involved in a finite group G satisfy the `inductive Feit condition', then Walter Feit's conjecture from 1980 holds for G. In particular, this would solve Brauer's Problem 41 from 1963 in the affirmative. This inductive Feit condition implies that some features of all the irreducible characters of finite groups can be found locally.

The Gluck–Wolf theorem and its general version [Navarro and Tiep, Annals of Math. 178 (2013), 1135–1171] relate arithmetic properties at a fixed prime of the ratios , for irreducible characters of a finite group that lie over a fixed ‐invariant irreducible character of a normal subgroup of , to the structure of Sylow ‐subgroups of . This result constituted a key step towards the recent proof [Malle et al., Annals of Math. 200 (2024), 557–608] of Brauer's Height Zero Conjecture. In this paper, we prove a further extension of the Gluck–Wolf theorem to sets of primes, with a mild condition on if the alternating group is involved in the group.

A new conjecture on characters of finite groups, related to the McKay conjecture, was proposed recently by the first and third authors. In this paper, we prove it for $p$-solvable groups when $p$ is odd.

If p and q are two primes, G is a finite group, and Bp(G) is the set of complex irreducible characters in the principal p-block of G, we study |Bp(G) boolean AND Bq(G)| and its possible relation with some local subgroup.

We determine the finite groups whose inequivalent real irreducible representations have different degrees.

This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(χ)$ of an irreducible character $χ$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $χ$ of $G$, we introduce an integer-valued invariant $S(G,χ,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $χ$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $χ$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,χ,n)$ is always non-negative, and it is positive if and only if a representation affording $χ$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $χ$ if and only if $S(G,χ, c(χ))>0$.

We study geometric monodromy groups $G_{\geo,\sF_q}$ of the local systems $\sF_q$ on the affine line over $\F_2$ of rank $D=\sqrt{q}(q-1)$, $q=2^{2n+1}$, constructed in \cite{Ka-ERS}. The main result of the paper shows that $G_{\geo,\sF_q}$ is either the Suzuki simple group $\tw2 B_2(q)$, or the special linear group $\SL_D$. We also show that $\sF_8$ has geometric monodromy group $\tw2B_2(8)$, and arithmetic monodromy group $\Aut(\tw2 B_2(8))$ over $\F_2$, thus establishing \cite[Conjecture 2.2]{Ka-ERS} in full in the case $q=8$.