# 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, $12^{\it th}$ 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.167–176
Title Scott-Wiegold conjecture ScottWiegoldConjecture 2013-03-22 18:29:34 2013-03-22 18:29:34 whm22 (2009) whm22 (2009) 8 whm22 (2009) Theorem msc 20E06 one relator products of cyclic groups