Related papers: It\^{o}'s theorem and monomial Brauer characters

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The…

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let S be the Steinberg representation of G(K). Let t be be a very regular semisimple element of G(K). In this paper we give a…

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $\pi$ be a set of primes, and let $G$ be a finite $\pi$-separable group. We consider the Isaacs ${\rm B}_\pi$-characters. We show that if $N$ is a normal subgroup of $G$, then ${\rm B}_\pi (G/N) = {\rm Irr} (G/N) \cap {\rm B}_\pi (G)$.

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either…

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup…

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$, or more generally, which representations remain homogeneous in characteristic $p$. In this paper we…

Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…

We construct, for every prime p, a function field K of characteristic p and an ordinary abelian variety A over K, with no isotrivial factors, that admits an etale self-isogeny of p-power degree. As a consequence, we deduce that there exist…

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the complement…

Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let \pi be an irreducible unitary representation of M and \phi (not necessarily…

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…

Consider a finite group $G$ of order $n$ with a prime divisor $p$. In this article, we establish, among other results, that if the Sylow $p$-subgroup of $G$ is neither cyclic nor generalized quaternion, then there exists a bijection $f$…

A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a…

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

Let $p \geq 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing…

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime…

Let G be a reductive group over a non-archimedean local field k. We provide necessary conditions and sufficient conditions for all tori of G to split over a tamely ramified extension of k. We then show the existence of good semisimple…