Two words *u* and *v* commute (that is, *uv=vu*) if and only if they have the same primitive root. This means that there is a primitive word *t* and integers *i,j* such that *u=t*^{i} and *u=t*^{j}.

This is the most simple form of the following stronger statement:

- Two words commute if and only if they satisfy a nontrivial relation. This, in turn, is the GraphLemma for two words.

Yet more general is the following fact which is connected to the PeriodicityLemma:

- Let
*w*and_{u}*w*be two words generated by_{v}*u*and*v*, where*w*starts with_{u}*u*and*w*starts with_{v}*v*. If*w*and_{u}*w*have a common prefix of length at least_{v}*|u|+|v|-GCD(|u|,|v|)*, then*u*and*v*commute.

-- StepanHolub - 10 Sep 2011

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Topic revision: r1 - 2011-09-10 - StepanHolub

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