Computing the Schur multipliers of the Lie \(p\)-rings in the family defined by a symbolic Lie \(p\)-ring presentation. (English) Zbl 1535.17021

Eick, Bettina; Ghorbanzadeh, Taleea Jalaeeyan

J. Symb. Comput. 106, 68-77 (2021).

Summary: A symbolic Lie \(p\)-ring presentation defines a family of nilpotent Lie rings with \(p^n\) 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 \(p^n\) for all primes \(p\) and \(n\leq 7\). 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 \(p^6\) for all primes \(p\geq 5\). Via the Lazard correspondence this yields the Schur multipliers of all groups of order dividing \(p^6\) for all primes \(p\geq 5\).

MSC:

17B50	Modular Lie (super)algebras
17B56	Cohomology of Lie (super)algebras
20D15	Finite nilpotent groups, \(p\)-groups
20C25	Projective representations and multipliers

Keywords:

Schur multiplier; Lie ring; \(p\)-group

Software:

GAP; LiePRing

References:

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