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, Aut P 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 algebras. These algebras also produce useful criteria for a p-group to be centrally indecomposable.