Decomposing p-groups via Jordan algebras

TL;DR

This paper explores the structure of finite p-groups of class 2 and exponent p by decomposing them using Jordan algebras, revealing invariants and automorphism properties related to their central decompositions.

Contribution

It introduces a novel approach using Jordan algebras to analyze p-group decompositions, providing new invariants and criteria for indecomposability.

Findings

Invariants include the number of members, orders of members, and orders of centers in decompositions.

Aut P is not always transitive on the set of decompositions, with the number of orbits being arbitrary.

Jordan algebra structure helps determine when a p-group is centrally indecomposable.

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