A Pastiche on Embeddings into Simple Groups (following P. E. Schupp)

Let λ be an infinite cardinal number and let C = {Hi | i ∈ I} be a family of nontrivial groups. Assume that 2 ≤ |I| ≤ λ, |Hi| ≤ λ, for i ∈ I, and at least one member of C achieves the cardinality λ. We show that there exists a simple group S of cardinality λ that contains an isomorphic copy of each member of C and, for all Hi, Hi′ in C with |Hi′ | = λ, is generated by the copies of Hi and Hi′ 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. Olshanskĭı. 1. Background and results In [Sch76] Schupp used small cancelation theory (construction of Adian-Rabin type) to prove, among other things, the following result. Theorem S (Schupp [Sch76]). Let G, H and K be nontrivial groups with |G| ≤ |H ∗K| and |K| ≥ 3. There exists a simple group S that contains an isomorphic copy of G and is generated by isomorphic copies of H and K. Corollary S. Let G be a countable group. For all p, q ∈ {2, 3, . . .} ∪ {∞} with q ≥ 3, there exists a simple group S that contains an isomorphic copy of G and is generated by a pair of elements of order p and q, respectively. The simple group constructed by Schupp in Theorem S, in addition to being dependent on G, depends on H and K. Accordingly, the simple group in Corollary S, in addition to being dependent on G, depends on the pair (p, q). We will show that the argument used by Schupp can be adapted in such a way that the same simple group can be used even if one considerably varies H and K in Theorem S and, consequently, the same simple group can be used independently of the pair (p, q) in Corollary S. Theorem A. Let |I| ≥ 2 and C = {Hi | i ∈ I} be a countable family of countable nontrivial groups, at least one of which has at least 3 elements (the groups may be isomorphic for different values of the index). There exists a 2-generated simple group S that contains an isomorphic copy of each member of C and, for all Hi, Hj in C with |Hj | ≥ 3, is generated by the copies of Hi and Hj in S. 2000 Mathematics Subject Classification. 20F06, 20E32.