### 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

Howie, J. (2002). A proof of the Scott-Wiegold conjecture on free products of cyclic groups.

*Journal of Pure and Applied Algebra*,*173*(2), 167-176. https://doi.org/10.1016/S0022-4049(02)00042-7