# Scott-Wiegold conjecture

The Scott-Wiegold conjecture (1976) is stated as follows:

Given distinct prime numbers^{} $p$, $q$ and $r$, the free product of cyclic groups^{} ${C}_{p}*{C}_{q}*{C}_{r}$ is not the normal closure^{} of any single element.

In 1992 this was included as problem 5.53 of The Kourovka Notebook: Unsolved Problems in [1].

The conjecture was proven to be true in 2001 by James Howie [2]. Despite remaining an unsolved problem for 25 years, the proof is both brief and fairly elementary.

Whilst the question is group theoretic and involves only , the proof does not use any combinatorial but instead depends on basic notions from topology.

## References

- 1 V.D.Mazurov, E.I. Khukhro (Eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, ${12}^{\mathrm{\u0e50\x9d\x91\u0e01\u0e42\x84\x8e}}$ Edition, Russian Academy of Sciences, Novosibirsk, 1992.
- 2 James Howie, A proof of the Scott-Wiegold conjecture on free products of cyclic groups, Journal of Pure and Applied Algebra 173, 2002 pp.167โ176

Title | Scott-Wiegold conjecture |
---|---|

Canonical name | ScottWiegoldConjecture |

Date of creation | 2013-03-22 18:29:34 |

Last modified on | 2013-03-22 18:29:34 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 8 |

Author | whm22 (2009) |

Entry type | Theorem |

Classification | msc 20E06 |

Synonym | one relator products of cyclic groups |