Finding central decompositions of p-groups

Abstract A Las Vegas polynomial-time algorithm is given to find a central decomposition of maximum size for a finite p-group of class 2. The proof introduces an associative *-ring as a tool for studying central products of p-groups. This technique leads to a translation of the problem into classical linear algebra which can be solved by application of the MeatAxe and other established module-theoretic algorithms. When p is small, our algorithm runs in deterministic polynomial time.