Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$-Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

We prove that a finite coprime linear group $G$ in characteristic $p\geq \frac{1}{2}(|G|-1)$ has a regular orbit. This bound on $p$ is best possible. We also give an application to blocks with abelian defect groups.

Bergelson and Tao have recently proved that if G is a D-quasi-random group, and x, g are drawn uniformly and independently from G, then the quadruple (g, x, gx, xg) is roughly equidistributed in the subset of G4 defined by the constraint that the last two coordinates lie in the same conjugacy class. Their proof gives only a qualitative version of this result. The present note gives a rather more elementary proof which improves this to an explicit polynomial bound in D−1.

Let Ω be a finite set and let G be a permutation group acting on it. A subset H of G is called t-intersecting if any two elements in H agree on at least t points. Let SDn and SBn be the classical Coxeter group of type Dn and type Bn, respectively. We show that the maximum-sized (2t)-intersecting families in SDn and SBn are precisely cosets of stabilizers of t points in [n] ≔ {1, 2, …, n}, provided n is sufficiently large depending on t.

We present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of . The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.