Related papers: New refinements of the McKay conjecture for arbitr…

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…

We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay. The conjecture was…

Let p be a prime, B a p-block of a finite group G and b its Brauer correspondent. According to the Alperin-McKay Conjecture, there exists a bijection between the set of irreducible ordinary characters of height zero of B and those of b. In…

Let $P$ be a Sylow $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime. Using a normal $p$-series $\mathcal{N}$ of $G$, we introduce the notion of $(\mathcal{N},p)$-stable characters and prove that $G$ and ${\bf N}_G(P)$…

Let $N$ be normal subgroup of a finite group $G$, $p$ be a prime, $P$ be a Sylow $p$-subgroup of $G$ and $\theta$ be a $P$-invariant irreducible character of $N$. Suppose that $G/N$ is a $p$-solvable group. In this note we show that,…

Let $G$ be a finite group, $p$ a prime number and $P$ a Sylow $p$-subgroup of $G$. Recently, G. Malle, G. Navarro, and P. H. Tiep conjectured that the number of $p$-Brauer characters of $G$ coincides with that of the normaliser ${\bf…

The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to $p$ ($p$ a prime) of a finite group $G$ and those of the subgroup $N$, the normalizer of…

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…

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets…

This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view,…

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…

We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and…

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…

Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay…

We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of H\'{e}thelyi and K\"{u}lshammer for…

We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$.…

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This…

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary…

In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results…