Radical subgroups and the inductive blockwise Alperin weight conditions for $\mathrm{PSp}_4(q)$

Abstract

We determine explicitly the $2$-radical subgroups and their normalizers for the group ${Sp}_4\left(q\right)$, where $q$ is odd. We then show that the corresponding simple group ${PSp}_4\left(q\right)$ satisfies the inductive blockwise Alperin weight conditions for the prime $2$ and odd primes dividing $q^2-1$. When combined with existing literature, this completes the verification that ${PSp}_4\left(q\right)$ satisfies the conditions for all primes and all choices of $q$.