Computing the Schur multipliers of the Lie p-rings in the family defined by a symbolic Lie p-ring presentation

Abstract

A symbolic Lie p-ring presentation defines a family of nilpotent Lie rings with pn elements for infinitely many primes p and a fixed positive integer n. Symbolic Lie p-ring presentations are used in the classification of isomorphism types of nilpotent Lie rings of order pn for all primes p and n≤7. We describe an algorithm to compute the Schur multipliers of all nilpotent Lie rings in the family defined by a symbolic Lie p-ring presentation. We apply this to determine the Schur multipliers of all nilpotent Lie rings of order dividing p6 for all primes p≥5. Via the Lazard correspondence this yields the Schur multipliers of all groups of order dividing p6 for all primes p≥5.