On a problem of kurosh, jonsson groups, and applications

We prove some results in group theory in a model theoretic spirit. (i) We construct Jonsson groups of cardinality χ, and other cardinalities as well. This answers an old question of Kurosh. (ii) Our group is simple with no maximal subgroup; so it follows that taking Frattini subgroups does not commute with direct products. (ii) Assuming the continuum hypothesis, our group is not a topological group, except with the trivial topologies. This answers a quite old question of A.A. Markov. In the construction we use small cancellation theory. We try to make the paper intelligible to both group theorists and model theorists. Only a knowledge of naive set theory and group theory is needed.