4 problems

Let $G$ be a $p$-solvable finite group and let $A$ be a $p$-subgroup contained in a unique Sylow $p$-subgroup of $G$. Thompson's conjecture. The $p$-length of $G$ is bounded in ter…

Let $p>2$, let $G$ be a $p$-solvable group, and let $\varphi\in\operatorname{IBr}(G)$ have vertex $Q$. For a vertex pair $(Q,\delta)$, let $L_\varphi(Q,\delta)$ denote the set of l…

Let $G$ be a finite group, let $p$ be a prime, let $P$ be a Sylow $p$-subgroup of $G$, and let $\chi\in\operatorname{Irr}(G)$. For a subgroup $Q\le P$, write $\chi(1)_p$ for the…

Let $p$ be a prime, let $G$ be a $p$-solvable finite group, and let $P$ be a Sylow $p$-subgroup of $G$. Let $\operatorname{N}_{G}(P)$ be the normalizer of $P$, and let…