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=ti and u=tj.

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 wu and wv be two words generated by u and v, where wu starts with u and wv starts with v. If wu and wv have a common prefix of length at least |u|+|v|-GCD(|u|,|v|), then u and v commute.

-- StepanHolub - 10 Sep 2011

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