On Groups Without Large Corefree Subgroups
On Groups Without Large Corefree Subgroups
L. Babai, A. J. Goodman, L. Pyber:
On Groups Without Large Corefree Subgroups
A subgroup H of a group G is 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.