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

TL;DR

This paper constructs large simple groups containing isomorphic copies of a given family of groups, extending Schupp's and Ol'shanskii's embedding results using small cancellation techniques.

Contribution

It generalizes Schupp's embedding theorem to uncountable groups and provides a new construction method for simple groups with specified subgroup embeddings.

Findings

Existence of simple groups of any given uncountable cardinality containing specified groups

Extension of Schupp's embedding results to uncountable cases

Partial recovery of Ol'shanskii's embedding results in the countable case

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.