Texas A&M University, Department of
Mathematics, 216 Milner Hall, 30th of January 2008, 3:00-3:50
Groups and Dynamics Seminar
Embeddings into
simple groups
Zoran Šunić of Texas A&M
University
We show that any countable family C of nontrivial countable groups can
be embedded into a 2-generated simple group S in such a way that any
two members of the family C (other than two copies of the cyclic
group of order 2) generate the group S.
A version of the result that does not use the countability assumption
is also provided. Let lambda be an infinite cardinal number and let C =
{H_i| i in I} be a
family of nontrivial groups. Assume that |I|<=lambda, |H_i|<=
lambda, for i in
I, and at least one member of C achieves the cardinality lambda. We
show that there exists a simple group S of cardinality lambda that
contains an isomorphic copy of each member of C and, for all H_i, H_j
in C with
|H_j|=lambda, is generated by the copies of H_i and H_j in S.
This generalizes a result of Paul E. Schupp (moreover, our proof
follows the
same approach based on small cancelation on free products). In the
countable case, we partially
recover a much deeper embedding result of Alexander Yu. Ol'shanskii.