Symbolic computation of Schur multipliers with an application to the groups of order dividing $p^6$

TL;DR

This paper presents an algorithm for computing Schur multipliers of nilpotent Lie p-rings, enabling the analysis of p-groups of order up to p^6, which advances understanding of their structure.

Contribution

The paper introduces a symbolic computation algorithm for Schur multipliers of nilpotent Lie p-rings, applicable to p-groups of order up to p^6.

Findings

Computed Schur multipliers for all p-groups of order dividing p^6

Demonstrated the algorithm's effectiveness on a broad class of nilpotent Lie p-rings

Enhanced understanding of the structure of p-groups of small order

Abstract

We describe an algorithm to compute the Schur multipliers of all nilpotent Lie $p$-rings in the family defined by a symbolic nilpotent Lie $p$-ring. Symbolic nilpotent Lie $p$-rings can be used to describe the isomorphism types of $p$-groups of order $p^n$ for $n \leq 7$ and all primes $p \geq n$. We apply our algorithm to compute the Schur multipliers of all $p$-groups of order dividing $p^6$.