Decomposing p-groups via Jordan algebras

By James B. Wilson. 2007

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Abstract 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 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.

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