A pastiche on embeddings into simple groups (following P. E. Schupp)

Abstract

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). In the countable case, we partially recover a much deeper embedding result of Alexander Yu. Ol'shanskii.