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We give new evidences to the fact that the structure of a solvable group can be controlled by irreducible monomial characters. In particular we inspect the role of monomial characters in Isaacs-Navarro-Wolf's conjecture and in Gluck's…

We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also…

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow $2$-subgroup. We generalize this result for Sylow $p$-subgroups, for any prime number $p$, while assuming the…

Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters in $B$ is $5$ then $D\cong C_5, C_7, D_8$ or $Q_8$, assuming that the…

Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture holds for the alternating groups when the prime p is odd.

We construct the supercharacter theory for the finite groups of triangular type. Its special case is the supercharacter theory for algebra groups of P.Diaconis and I.M.Isaacs. The supercharacter analog of the A.A. Kirillov formula for…

This paper is concerned with the representation theory of finite groups. According to Robinson, the truth of certain variants of Alperin's weight conjecture on the $p$-blocks of a finite group would imply some arithmetical conditions on the…

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in…

By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

The notion of the supercharacter theory was introduced by P.Diaconis and I.M.Isaaks in 2008. In this paper we review the main statements of the general theory, we observe the construction of supercharacter theory for algebra groups and the…

Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a "large" orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This extends (at the cost of a weaker bound) a 2005…

We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that…

The Alperin--McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its $p$-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this…

We prove that the p-Quillen complex of a finite solvable group with cyclic derived group is Cohen-Macaulay, if p is an odd prime. If p = 2 we prove a similar conclusion, but there is a discussion to be made.

Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by the second author. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower…