10.49 (1986)

Does there exist a group $G$ satisfying the following four conditions:
$\qquad$ 1) $G$ is simple, moreover, there is an integer $n$ such that $G = C^n$ for any conjugacy class $C$,
$\qquad$ 2) all maximal abelian subgroups of $G$ are conjugate in $G$,
$\qquad$ 3) every maximal abelian subgroup of $G$ is self-normalizing and it is the centralizer of any of its nontrivial elements,
$\qquad$ 4) there is an integer $m$ such that if $H$ is a maximal abelian subgroup of $G$ and $a \in G \setminus H$ then every element in $G$ is a product of $m$ elements in $aH$?