3 years ago

# Free subgroups in maximal subgroups of

Given a non-commutative finite-dimensional $F$-central division ring $D$, $N$ a subnormal subgroup of $GLn\left(D\right)$ and $M$ a non-abelian maximal subgroup of $N$, then either $M$ contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup $A$ of $M$ such that $K=F\left[A\right]$ is a subfield of $Mn\left(D\right)$, $K/F$ is Galois and $Gal\left(K/F\right)\cong M/\left(K\ast \cap N\right)$, also $Gal\left(K/F\right)$ is a finite simple group with $F\left[M\right]=Mn\left(D\right)$.