On conjectures of Olsson, Brauer, and Alperin

Let $G$ be a finite group, $p$ a prime number, $B$ a $p$-block of the group $G$, $k(B)$ the number of irreducible complex characters of $R$ belonging to $B$, $k_0(B)$ the number of irreducible characters of height zero in $B$, and let $D$ be the defect group of $B$. This article considers the relationship between Brauer's conjecture ($k(B)\leqslant|D|$), Olsson's conjecture ($k_0(B)\leqslant|D/D'|$), and Alperin's conjecture ($k_0(B)=k_0(\widetilde{B}$, where $\widetilde{B}$ is a $p$-block $N_G(D)$ such that $\widetilde{B}^G=B$). In particular, Olsson's conjecture is proved for $p$-blocks for those $p$-solvable groups $G$ for which a Hall $p'$-subgroup of the group $N_G(D)$ is either supersolvable or has odd order.