### Abstract

Problem 5.53 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 12th Edition, Russian Academy of Sciences, Novosibirsk, 1992) (contributed by Wiegold, attributed to Scott) asks whether a free product of three (finite) cyclic groups can be normally generated by a single element. We give a proof of the conjectured negative answer, and an application to Dehn surgery on knots: if Dehn surgery on a knot is S^{3} gives a connected sum, then all but at most 2 of the connected summands are Z-homology spheres, and hence (by a result of Valdez and Sayari) the number of connected summands is at most 3. © 2002 Published by Elsevier Science B.V.

Original language | English |
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Pages (from-to) | 167-176 |

Number of pages | 10 |

Journal | Journal of Pure and Applied Algebra |

Volume | 173 |

Issue number | 2 |

DOIs | |

Publication status | Published - 24 Aug 2002 |

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### Cite this

*Journal of Pure and Applied Algebra*,

*173*(2), 167-176. https://doi.org/10.1016/S0022-4049(02)00042-7

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*Journal of Pure and Applied Algebra*, vol. 173, no. 2, pp. 167-176. https://doi.org/10.1016/S0022-4049(02)00042-7

**A proof of the Scott-Wiegold conjecture on free products of cyclic groups.** / Howie, James.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A proof of the Scott-Wiegold conjecture on free products of cyclic groups

AU - Howie, James

PY - 2002/8/24

Y1 - 2002/8/24

N2 - Problem 5.53 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 12th Edition, Russian Academy of Sciences, Novosibirsk, 1992) (contributed by Wiegold, attributed to Scott) asks whether a free product of three (finite) cyclic groups can be normally generated by a single element. We give a proof of the conjectured negative answer, and an application to Dehn surgery on knots: if Dehn surgery on a knot is S3 gives a connected sum, then all but at most 2 of the connected summands are Z-homology spheres, and hence (by a result of Valdez and Sayari) the number of connected summands is at most 3. © 2002 Published by Elsevier Science B.V.

AB - Problem 5.53 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 12th Edition, Russian Academy of Sciences, Novosibirsk, 1992) (contributed by Wiegold, attributed to Scott) asks whether a free product of three (finite) cyclic groups can be normally generated by a single element. We give a proof of the conjectured negative answer, and an application to Dehn surgery on knots: if Dehn surgery on a knot is S3 gives a connected sum, then all but at most 2 of the connected summands are Z-homology spheres, and hence (by a result of Valdez and Sayari) the number of connected summands is at most 3. © 2002 Published by Elsevier Science B.V.

UR - http://www.scopus.com/inward/record.url?scp=0037167295&partnerID=8YFLogxK

U2 - 10.1016/S0022-4049(02)00042-7

DO - 10.1016/S0022-4049(02)00042-7

M3 - Article

VL - 173

SP - 167

EP - 176

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 2

ER -