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I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for it but didn't seem to find anything relevant. So is this article available online? If so, can you point me to it? In any case, can you give me a couple of examples of presentations of that group?

MickG
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  • The simplest example is of course $G=\langle \rangle$ :) – Hagen von Eitzen Jul 23 '15 at 14:49
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    http://www.ms.unimelb.edu.au/~cfm/papers/paperpdfs/cfmpes990426.pdf http://mathoverflow.net/questions/45798/presentations-of-the-trivial-group http://journals.cambridge.org/article_S0004972700018529 This is still not Neumann, but you should have used trivial group instead of identity group as keyword in google – Hagen von Eitzen Jul 23 '15 at 14:54
  • @HagenvonEitzen I look at the calendar - no, it's not April 1. Surely nobody actually wrote a paper on this...? – David C. Ullrich Jul 23 '15 at 16:20
  • I suspect you might be looking for this construction but I have not been able to find where Neumann initially used it. –  Jul 23 '15 at 16:35
  • Ah yes, I see have just repeated my answer to the earlier question! – Derek Holt Jul 23 '15 at 16:37
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    By the way, the trivial group is not a simple group (for the same sort of reason that $1$ is not a prime). – Derek Holt Jul 23 '15 at 16:37
  • @DerekHolt I didn't mean simple in the mathematical sense of the word, but in the common speech sense :). – MickG Jul 23 '15 at 17:12
  • @DavidC.Ullrich that is what my teacher said. And no, it wasn't April 1st then -- it was November 11th. The exact sentence: «In particolare per H = 1 c’è un articolo di Bernard Neumann che mostra come ideare presentazioni del gruppo identico G = {1} e le variazioni di queste presentazioni, cioè il numero di operazioni che ci vogliono per stabilire che è il gruppo identico.», which translates to [continues below] – MickG Jul 23 '15 at 17:19
  • «In particular, for H = 1 there is an article by Bernard Neumann that shows how to design presentations of the identical group G = {1} and the variations of these presentations, that is, the number of operations that are needed to establish it is the identical group.». PS the typing mistake is my bad: I had no idea it was "Bernhard", with that h in the middle. – MickG Jul 23 '15 at 17:19

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One such example is $\langle a,b,c \mid a^{-1}ba=b^2, b^{-1}cb=c^2, c^{-1}ac=a^2 \rangle$.

You can use this to construct a sequence of examples of increasing complexity. The example above is the first in the sequence and has total relator length $15$. The second group in the sequence is

$$\langle a,b,c \mid A^{-1}BA=B^2, B^{-1}CB=C^2, C^{-1}AC=A^2 \rangle,$$ where $A=a^{-1}bab^{-2}$, $B=b^{-1}cbc^{-2}$, $C=c^{-1}aca^{-2}$, so the total relator length is $75$. You can then repeat this idea to get further more complicated examples.

If you believe that the first presentation defines the trivial group, then it is not hard to prove that the second one does too. The first group is easily proved trivial by coset enumeration programs, but the second one is much harder.

Derek Holt
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  • Do you know where this class of examples came from? I have seen it attributed to Neumann a couple of places, but I am not sure where he initially showed this class of examples. –  Jul 23 '15 at 16:41
  • I have found a reference to B.H. Neumann, `Proofs', Math. Intelligencer 2 (1979), 18-19. This should be in our department library so I can check it tomorrow. – Derek Holt Jul 23 '15 at 17:32