Decomposing p-groups via Jordan algebras

For finite p-groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P: the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, AutP is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semi-simple and radical structure of Jordan rings. These rings also produce useful criteria for a p-group to be centrally indecomposable. Applications to p-groups of class 2 and arbitrary exponent are also provided.