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July 2011 - December 2015

## Publications

Publications (31)

In this paper we complete the proof of Brauer’s height zero conjecture for two primes proposed by G. Malle and G. Navarro.

Let p,q be different primes and suppose that the principal p- and the principal q-block of a finite group have only one irreducible complex character in common, namely the trivial one. We conjecture that this condition implies the existence of a nilpotent Hall {p,q}-subgroup and prove that a minimal counter-example must be an almost simple group wh...

In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z(G) isomorphic to A6,A7 or a simple group of Lie type with an exception...

Geoffrey Robinson conjectured in 1996 that the $p$ -part of character degrees in a $p$ -block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then s...

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ℓ > 5 no ℓ-block of any quasi-simple group can be a minimal counterex...

The Hilbert divisor pa(φ) of an irreducible p-Brauer character φ of a finite group G carries deep information about φ, respectively the module affording φ. In [8] we conjectured that φ belongs to a p-block of defect 0 if and only if its Hilbert divisor is 1. In this note we continue our investigations.

Robinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks o...

We study p-Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p-singular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character φ there exists a minimal p-power, say pa(φ), such that pa(φ)φ is quasi-...

This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group G, whose vertices are the prime divisors of |G| and there is an edge between two vertices p \ne q if and only if the principal p- and q-blocks of G have a nontrivial common complex irreducible ch...

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2...

Let G be a finite group, and write cd(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{cd}(G)$$\end{document} for the degree set of the complex irreducible cha...

Let G be a finite group and p be a prime. In this note, we show that if Op(G) = 1 and all subgroups of G of order p are conjugate, then either G has a p-block of defect zero, or p = 2 and G is a direct product of a simple group S {M22,A7} and an odd order group. This improves one of our previous works.

In this note, it is shown that a finite group G is solvable if for each odd prime divisor p of , , where is the set of complex irreducible characters of the principal p-block of G. Also, the structure of such groups is investigated. Examples show that the bound 2 is best possible.

In this note, we characterize finite nonsolvable groups whose principal p-blocks consist of ordinary irreducible characters of prime power degrees for every prime p. In addition, we give an upper bound of the nilpotent length for the solvable situation.

The main purpose of this note is to show that there is a one-to-one correspondence between minimal non-nilpotent (resp., locally nilpotent) saturated fusion systems and finite p′-core-free p-constrained minimal non-nilpotent (resp., locally p-nilpotent) groups. © 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzh...

Let G be a finite group, and write (Formula presented.) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees (Formula presented.), the total number of (not necessarily different) primes of the greatest common divisor (Formula presented.) is at most 2. In...

Recently, Isaacs, Moretó, Navarro, and Tiep investigated finite groups with just one irreducible character degree divisible by a given prime p, and showed that their Sylow p-subgroups are almost normal and almost abelian. In this paper, we consider the corresponding situation for Brauer characters. In particular, we show that if a finite group G ha...

A finite simple group of Lie type in defining characteristic p has exactly two p-blocks, the principal block and a block of defect zero consisting of the Steinberg character whose degree is the p-part of the order of the group. In this paper we characterize finite groups G which have exactly the principal p-block and a p-block of defect zero consis...

In this paper we characterize finite quasi-simple groups of

Let
$p$
be an odd prime. In this note, we show that a finite group
$G$
is solvable if all degrees of irreducible complex characters of
$G$
not divisible by
$p$
are either 1 or a prime.

Let p be a prime. The Sylow p-number of a finite group G, which is the number of Sylow p-subgroups of G, is called solvable if its ℓ-part is congruent to 1 modulo p for any prime ℓ. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow p-number of a finite group is the product of two kinds of factor...

Motivated by Isaacs and Passman’s characterization of finite groups all of whose nonlinear complex irreducible characters have prime degrees, we investigate finite groups [Formula: see text] with exactly one character degree that is not a prime. We show that either [Formula: see text] is solvable with [Formula: see text] or [Formula: see text] for...

In this note, we show that if a finite group G has only one conjugacy class of subgroups of an odd prime order p, then G has a p-block of defect zero if and only if O
p
(G) = 1. Generally, this result is not true for p even. However, we show that it is true if G does not have quotient groups isomorphic to M
22 or A
7.

Let
$G$
be a finite group and
$\mathrm{bcl}(G)$
the largest conjugacy class length of
$G$
. In this note we slightly improve He and Shi’s lower bound for
$\mathrm{bcl}(G)$
, showing that
$|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$
.

Let G be a finite group with the degree set cd(G)cd(G) of its complex irreducible characters. We call that G satisfies the prime-power hypothesis if, for distinct degrees χ(1),ψ(1)∈cd(G)χ(1),ψ(1)∈cd(G), the greatest common divisor gcd(χ(1),ψ(1))gcd(χ(1),ψ(1)) is a prime power. In this paper, we show that |cd(G)|⩽18|cd(G)|⩽18 if G is a nonsolvable g...

The main purpose is to investigate almost simple DD-groups. It proves that the almost simple group G is not a DD-group unless G is one of the following groups: 1) M22, J2, Co1, Fi'24, McL, Th, B, and the automorphism group of M12 or J2; 2) A5, A6, A7, A9, A10, A16, S5, Aut(A6), S8, S10, or An (62≤n≤205); 3) L3(2), Aut(L3(3)), or L2(q) for q=4, 5, 7...

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol).

Our main purpose of this paper is to give π-block forms of Brauer’s k(B) −conjecture and Olsson’s conjecture for finite π −separable groups.
Keywords
π-blocks–Olsson’s conjecture–
π-separable groups

In this paper, we show that the alternating group A5 is the only nonsolvable group whose character graph has no triangles.

In this note, we first investigate the degrees of irreducible π-Brauer characters of a finite π-separable group G. Then we relate degrees of irreducible π-Brauer characters to class lengths of π-regular elements of G.