Group relations coming from English words having same pronunciation
Let us check in GAP the result of the paper on group relations coming from English words having same pronunciation.
This paper is called “Quotients homophones des groupes libres” (Homophonic quotients of free groups) and is written by Jean-François Mestre, René Schoof, Lawrence Washington and Don Zagier. It was published in Experiment. Math. 2 (1993), no. 3, 153–155. The authors take the free group on 26 letters a, b, c, …, z and consider its two quotients (denote them G and H) by the relations of the form A=B where A and B are words having the same pronunciation in English and French languages respectively for G and H. The authors prove that both G and H are trivial.
The paper is written in two columns. The left column is written in French and proves the main theorem for the group defined by the relations coming from English (buy=by, sign=sine, etc.), while the right column is written in English and proves the same for the group coming from French. It even contains a reference to Lam’s book “Serre’s Conjecture” that is used to prove “the triviality of r from the well known fact serre=sert“.
I wrote a simple GAP script which checks that the statement of the paper is true for the group defined by the English language. You may find it on GitHub here: homophony.g. Just paste it into the GAP session and it will immediately confirm that the order of this group is 1. The script also demonstrates that the relation “ruff=rough“, used in the original paper, is in fact superfluous – GAP can determine the order of the group without using it.
The authors claim that their approach should be very useful to investigate the famous P=NP conjecture, so perhaps the next step should be checking with GAP that the group coming from French language is trivial…
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