Induction of Characters and Finite $p$-Groups

分支群的子群归纳性质

Subgroup induction property for branch groups

本文研究了分支群的子群归纳性质，证明了具有该性质的有限生成分支群必为挠群、刚好无限且子群可分，并证明所有挠GGS群均具有此性质，提供了首个无限族例子。

Solomon's induction in quasi-elementary groups

Given a finite group G, we address the following question: which multiples of the trivial representation are linear combinations of inductions of trivial representations from proper subgroups of G? By Solomon's induction theorem, all multiples are if G is not quasi-elementary. We complement this by showing that all multiples of p are if G is p-quasi-elementary and not cyclic, and that this is best possible.

Categorifying induction formulae via divergent series

We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.

Superinduction for pattern groups

It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies Diaconis--Isaacs' concept of superinduction in pattern groups. While superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by findi…

Norm formulas for finite groups and induction from elementary abelian subgroups

It is known that the norm map N_G for a finite group G acting on a ring R is surjective if and only if for every elementary abelian subgroup E of G the norm map N_E for E is surjective. Equivalently, there exists an element x_G in R with N_G(x_G) = 1 if and only for every elementary abelian subgroup E there exists an element x_E in R such that N_E(x_E) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for x_G in terms of the elements x_E. In this paper we present a method to solve this problem for an arbitrary group G and an arbitrary group action on a r…

The McKay conjecture and Brauer's induction theorem

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 propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broué abeli…