Scott-Wiegold conjecture

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

Given distinct prime numbersMathworldPlanetmath p, q and r, the free product of cyclic groupsMathworldPlanetmath Cp*Cq*Cr is not the normal closurePlanetmathPlanetmathPlanetmath 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.


  • 1 V.D.Mazurov, E.I. Khukhro (Eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 12๐‘กโ„Ž 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.167โ€“176
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