Computing \(p\)-groups with trivial Schur multiplicator.
type: Item
instance of: scholarly article
title: Computing \(p\)-groups with trivial Schur multiplicator.
zbMATH Open document ID: 1196.20019
author: B. Eick
published in: Journal of Algebra
publication date: 2009-10-02T00:00:00Z
review text: The author continues her investigation of \(p\)-groups with trivial Schur multiplicator. She has previously proved [Isr. J. Math. 166, 157-166 (2008; Zbl 1153.20015)] that, for odd primes, there are only finitely many \(p\)-groups with trivial Schur multiplicator of a given coclass. (Recall a \(p\)-group \(G\) of order \(p^n\) has coclass \(r\) if \(r=n-c\) and \(c\) is the nilpotency class of \(G\).) The author introduces an algorithm to compute these finitely many \(p\)-groups for a given prime and coclass. The author classifies all such groups of coclass 1, gives a conjectural description for them for coclass 2 and determines them for coclass 3 and \(p\in\{3,5,7\}\) and coclass 4 and \(p=3\). The algorithm computes Schur towers. A nilpotent group \(G\) is a Schur tower if every lower central series factor \(G/\gamma_{j+1}(G)\) is a Schur cover of the factor \(G/\gamma_j(G)\). Every nilpotent group with trivial Schur multiplicator is a Schur tower and furthermore, for odd \(p\), there are only finitely many Schur towers among the \(p\)-groups of coclass \(r\). A main part of the algorithm is an adaptation of an algorithm of \textit{W. Nickel} [PhD thesis, ANU, Canberra (1993)] which computes the isomorphism types of Schur covers of finite \(p\)-groups. The author is led to conjecture that, for \(p\) odd, the order of a \(p\)-group with trivial Schur multiplicator and coclass \(r\) is bounded by a function \(b(r)\) which is independent of \(p\). \textit{M. F. Newman} has confirmed this for coclass 2 [J. Algebra 322, No. 3, 910-913 (2009; Zbl 1196.20020)].
reviewed by: Rachel D. Camina
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 5610502
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Schur multiplicators
zbMATH Keywords: coclasses
zbMATH Keywords: Schur towers
zbMATH Keywords: Schur covers
zbMATH Keywords: algorithms
zbMATH Keywords: lower central series
describes a project that uses: AutPGrp
describes a project that uses: Polycyclic
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.03.031
OpenAlex ID: W2087573412
cites work: Computing with infinite polycyclic groups
cites work: On the Fitting subgroup of a polycyclic-by-finite group and its applications
cites work: Schur multiplicators of finite \(p\)-groups with fixed coclass.
cites work: CONSTRUCTING AUTOMORPHISM GROUPS OF p-GROUPS
cites work: Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group.
cites work: Q4705901
cites work: Zu einem von B. H. und H. Neumann gestellten Problem
cites work: Q4867135
cites work: Q4650358
cites work: A Property of Finite p-Groups with Trivial Multiplicator
cites work: Q4544915
cites work: Q4946828
cites work: On coclass and trivial Schur multiplicator.
cites work: The groups with order \(p^7\) for odd prime \(p\).
cites work: Q3961021
DOI: 10.1016/J.JALGEBRA.2009.03.031
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description: scientific article; zbMATH DE number 5610502