相关论文: Counting lifts of Brauer characters

Let $p$ be an odd prime, and suppose that $G$ is a $p$-solvable group and $\varphi\in {\rm IBr}(G)$ has vertex $Q$. In 2011, Cossey, Lewis and Navarro proved that the number of lifts of $\varphi$ is at most $|Q:Q'|$ whenever $Q$ is normal…

In 2007, J. P. Cossey conjectured that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. In this paper we revisited…

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup of a subgroup $V$. Suppose $\phi \in \ibr G$. If either $|G|$ is odd or $p = 2$, we prove that the number of Brauer characters of $H$ inducing $\phi$ with vertex…

If $G$ is a solvable group and $p$ is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character $\phi$ of $G$, there exists a character $\chi \in \irrg$ such that $\chi^o = \phi$, where $^o$ denotes the…

Navarro defined the set ${Irr}(G \mid Q, \delta) \subseteq {Irr}(G)$, where $Q$ is a $p$-subgroup of a $p$-solvable group $G$, and shows that if $\delta$ is the trivial character of $Q$, then ${Irr}(G \mid Q, \delta)$ provides a set of…

A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. This conjecture…

Suppose $G$ is a $p$-solvable group, where $p$ is odd. We explore the connection between lifts of Brauer characters of $G$ and certain local objects in $G$, called vertex pairs. We show that if $\chi$ is a lift, then the vertex pairs of…

The Fong-Swan theorem shows that for a $p$-solvable group $G$ and Brauer character $\phi \in \ibrg$, there is an ordinary character $\chi \in \irrg$ such that $\chi^0 = \phi$, where $^0$ denotes restriction to the $p$-regular elements of…

Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $\delta$ is a linear character of $Q$ with the property that $\delta(x)=\delta(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this…

Let $B$ be a $p$-block of a finite group $G$ with defect group $D$. The more difficult direction of the recently proven height zero conjecture says that $D$ is abelian if every character in Irr$(B)$ has height zero. We consider a smaller…

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1.…

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension…

In this paper, we consider lifts of $\pi$-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we…

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $\varphi$. We prove that $\varphi$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms 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…

Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ divides $\chi(1)$ for some $\chi \in IBr_p(G)$. We show that…

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 solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $\varphi(1)$ for every irreducible monomial $p$-Brauer character $\varphi$ of $G$ if and only if $G$ has a normal Sylow…

Let $p$ be a prime. We classify the finite groups having exactly two irreducible $p$-Brauer characters of degree larger than one. The case, where the finite groups have orders not divisible by $p$, was done by P. P\'alfy in 1981.

For a torsion unit $u$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$, and a prime $p$ which does not divide the order of $u$ (but the order of $G$), a relation between the partial augmentations of $u$ on the $p$-regular…