L\'aszl\'o Babai, Albert J. Goodman, L\'aszl\'o Pyber:

        On Groups Without Large Corefree Subgroups

A subgroup $H$ of a group $G$ is {\em corefree} if $H$ contains no
non-trivial normal subgroup of $G$, or equivalently the transitive
permutation representation of $G$ on the cosets of $H$ is faithful.
We call a subgroup $D$ a ``dedekind'' subgroup of $G$ if all subgroups
of $D$ are normal in $G$.  Our main result is the following: If a
finite group $G$ has no corefree subgroups of order greater than $k$, 
then $G$ has two dedekind subgroups $D_1$ and $D_2$ such that every
subgroup in $G$ of order greater than $f(k)$ has non-trivial
intersection with either $D_1$ or $D_2$ (where $f$ is a fixed function
independent of $G$). Examples show that the dedekind subgroups need
not have index bounded by a function of $k$, and the result would not
be true with one dedekind subgroup instead of two. We exhibit various
related properties of $p$-groups and locally finite groups without
large corefree subgroups, including the following: If $G$ is a locally
finite group with no infinite corefree subgroup, then every infinite
subgroup of $G$ contains a non-trivial cyclic normal subgroup of $G$.