# A normal-form theorem for monoids and groups with
the single relationxy<->yx

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0725.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**A normal-form theorem for monoids and groups with
the single relation xy<->yx**

The word problem for confluent Thue systems is linear-time
and for almost confluent systems it is
PSPACE-complete. Here we consider a single length-preserving
rule, of the form xy<->yx, whose word problem
could turn out to be tractable.\
A search for normal forms
leads to the conjecture that if x^m y^n <->\* x^p y^q
then m=p and n=q.

We prove a stronger version of this: in a group G with the
single relator xyx^{-1}y^{-1} where x
and y are non-commuting
, if x^my^{-n} = 1 in G then m=n=0.