On nilpotent groups on exponent p
The paper discusses amalgams of groups and applications to countable existentially closed groups. An amalgam of groups consists of a pair of groups A, B with a common subgroup D and a realization of this amalgam consists of a group C together with embeddings of A and B into C which co-incide on D. The general problem considered is to find such an amalgam with C suitable restricted relative to A and B. Let p be a prime. Let \({\mathbb{K}}\) denote the class of nilpotent groups of exponent p and class less than p. Let \({\mathbb{K}}'\) denote the class of nilpotent groups with central quotient of exponent p. Then criteria are given for an amalgam of groups in \({\mathbb{K}}\) or \({\mathbb{K}}'\) to be realizable in \({\mathbb{K}}\) or \({\mathbb{K}}'\). The author also describes the groups D in either of these two classes which have the property that any amalgam with common subgroup D can be realized within the class. The results are then applied to the existence and uniqueness of countable existentially closed groups within various classes. Amongst other things, it is shown that the class \({\mathbb{K}}\) has a unique countable existentially closed group and that the class of all torsion groups in \({\mathbb{K}}'\) has a unique countable existentially closed group.
scientific article; zbMATH DE number 4122225