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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2017: 0.32

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On torsion in finitely presented groups

Maurice Chiodo
Published Online: 2014-04-15 | DOI: https://doi.org/10.1515/gcc-2014-0001


We describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P, isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π20-hard, Σ20-hard, and lies in Σ30. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ20 sets which are closed under taking factors.

Keywords: Higman's embedding theorem; universal finitely presented group; embeddings; arithmetic hierarchy; torsion

MSC: 20F10; 03D40; 03D80

About the article

Received: 2013-10-03

Published Online: 2014-04-15

Published in Print: 2014-05-01

Funding Source: University of Melbourne Overseas Research Experience Scholarship

Funding Source: FIRB “Futuro in Ricerca”

Award identifier / Grant number: RBFR10DGUA_002

Funding Source: Swiss National Science Foundation

Award identifier / Grant number: FN PP00P2-144681/1

Citation Information: Groups Complexity Cryptology, Volume 6, Issue 1, Pages 1–8, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2014-0001.

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