In this paper, we consider lifts of $π$-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 call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such lifts.
Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $δ$ is a linear character of $Q$ with the property that $δ(x)=δ(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,δ)$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.
In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of $\Aut(P)$ lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough…
Parity patterns associated with lifts of Hecke groups
令$q$为奇素数，$m$为正整数，且令$\Ga_m(q)$为由两个元素$x$和$y$服从关系$x^{2m}=y^{qm}=1$和$x^2=y^q$生成的群;即，$\Ga_m(q)$为两个分别为$2m$和$qm$的循环群的自由积，沿着它们的$m$的子群合并。我们的主要结果确定了[T. W. Müller，Adv. in Math.153（2000），118-154]，并且在当$\gcd(m,| H|)=1$ .这个计算依赖于Hecke群$\mathfrak H(q)=C_2*C_q$中三个计数问题的解：（i）确定$\mathfrak H(q)$的子群数的奇偶性;（ii）确定与$C_2$和$C_\infty$的拷贝的自由积同构的$\mathfrak H(q)$的指数$n$子群数的奇偶性;（iii）确定$\mathfrak H(q)$中与$C_q$拷贝的自由产物同构的指数$n$子群的数目的奇偶性。第一个问题在[T. W. Müller，in：{ \it Groups：Topological，Combinatorial and Arithmetic Aspects}（T. W. Müller编），LMS Lecture Notes Series 311，Cambridge University Press，Cambridge，2004，pp. 327-374]。本文的大部分内容涉及问题（i…
The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this "Hypothesis H", here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman's comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility o…
We will describe the relationship between the indecomposable characters of the finitary symmetric group and its ergodic invariant random subgroups; and we will interpret each Thoma character as an asymptotic limit of a naturally associated sequence of characters induced from linear characters of Young subgroups of finite symmetric groups.
The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $Γ$ be a group and $Δ$ a subgroup of finite index. Suppose $Δ$ has normal subgroup growth of type $n^{\log n}$, does $Γ$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is …
A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $φ$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $φ$ is at most $|Q:Q'|$. This conjecture is now known to be true in several situations for $p$ odd, but there has been little progress for $p$ even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift are neither linear nor conjugate. In this paper we show that if $χ$ is a lift of an irreducible $2$-Brauer character in a solvable group, then $χ$ has a linear Navarro vertex …
Let $H$ be an abelian subgroup of a finite group $G$ and $π$ the set of prime divisors of $|H|$. We prove that $|H O_π(G)/ O_π(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is nilpotent.
We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents.
The character table of a finite group G determines whether |P:P'|=p^2 and whether |P:Z(P)|=p^2, where P is a Sylow p-subgroup of G. To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.
