21.121 (2026)

Let $p$ be a prime number. A group $\Gamma$ is called $p$-Jordan if there exist constants $J$ and $e$ such that any finite subgroup $G \subset \Gamma$ contains a normal abelian subgroup of order coprime to $p$ and of index at most $J \cdot |G_{(p)}|^e$. (For example by the results of Brauer–Feit and Larsen–Pink, for any field $K$ of characteristic $p$ the group $\text{GL}_n(K)$ is $p$-Jordan with $e = 3$.) Let the $p$-Jordan exponent $e(\Gamma)$ of the group $\Gamma$ be the infimum of all constants $e$ for which the above bound holds for some constant $J = J(e)$.
$\qquad$ a) Is it true that this infimum is always attained?
$\qquad$ b) Is it true that $e(\Gamma) \leqslant 3$ for any $p$-Jordan group $\Gamma$?