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Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.
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Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $mathrm{F}_4$, where we observe that the parametrization is uniform over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely many continuous ergodic invariant random subgroups.
In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.
For a finite group $G$, let $K(G)$ denote the field generated over $mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G. Thompson proved that $$K(A_n)=mathbb{Q}left ({ sqrt{p^*} : pleq n {text{ an odd prime with } p eq n-2}}right),$$ where $p^*:=(-1)^{frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $A_n$-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a $pi$-number is a positive integer whose prime factors belong to a set of odd primes $pi:= {p_1, p_2,dots, p_t}$. Let $K_{pi}(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $pi$-numbers. If $tgeq 2$, then we determine a constant $N_{pi}$ with the property that for all $n> N_{pi}$, we have $$K_{pi}(A_n)=mathbb{Q}left(sqrt{p_1^*}, sqrt{p_2^*},dots, sqrt{p_t^*}right).$$
In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups $mathrm{GL}_mleft(mathbb{F}_qright)$ and $mathrm{SL}_2left(mathbb{F}_qright)$.
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