4 problems

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Let $n$ be the number of irreducible characters of $G$ whose degrees are divisible by $p$…

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Write $\operatorname{Irr}_{p'}(G)$ for the set of irreducible characters of $G$ whose deg…

Let $A$ be an integral $k\times k$ matrix defining a linear system $L_3=A X^T$, where $X=(x_1,\ldots,x_k)^T$ and the component forms are $f_i(X)$. Say that $A$ has the good propert…

Let $f_1(x_1,\ldots,x_k),\ldots,f_s(x_1,\ldots,x_k)$ be multivariable polynomials with integral coefficients. Assume that there is a positive integer $c$ such that for every intege…