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Is there any non-trivial problem (maybe open problem) about the trivial group?

I asked already a question about the Laws characterizing the trivial group. There is a description of such laws. As another example, one can ask about the first order sentences characterizing the trivial group. I am not sure if these sort of problems are trivial or easy. Is there any important (open) question about the trivial group?

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    $\begingroup$ Google Andrews–Curtis conjecture $\endgroup$
    – Ievgen Bondarenko
    Commented Mar 1, 2017 at 7:34
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    $\begingroup$ @ HJRW, I didn't mean the first order theory. I meant the first order sentences which are true ONLY in the trivial group. The sentence $\forall x \ x=1$ is one example. Another example is $\forall\ x, y\ ( [x,y]=1\to x^3y^2=1)$. How can we characterize all these sentences? $\endgroup$
    – Sh.M1972
    Commented Mar 1, 2017 at 9:29
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    $\begingroup$ "Groups of order 1" by E. S. Rapaport $\endgroup$
    – bof
    Commented Mar 1, 2017 at 11:09
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    $\begingroup$ It is undecidable of a finite presentation presents the trivial group $\endgroup$
    – Benjamin Steinberg
    Commented Mar 1, 2017 at 13:08
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    $\begingroup$ Determining whether something is the trivial group is not a problem about the trivial group. $\endgroup$
    – YCor
    Commented Mar 1, 2017 at 19:13

2 Answers 2

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Problem 1.12 of [Unsolved Problems in Group Theory, The Kourovka Notebook, Novosibirsk, 2010]:

(W. Magnus) The problem of the isomorphism to the trivial group for all groups with $n$ generators and $n$ defining relations, where $n>2$.

Problem 1.13 of [Unsolved Problems in Group Theory, The Kourovka Notebook, Novosibirsk, 2010]:

(J. Stallings). If a finitely presented group is trivial, is it always possible to replace one of the defining words by a primitive element without chaning the group?

The answer is no, not always (S. V. Ivanov, Invent. Math., 165, no. 3 (2006), 525-549.

Problem 2.80 of 1 Does every non-trivial group satisfying the normalizer condition contain a non-trivial abelian normal subgroup? (S.N.Chernikov)

One may propose Problem 2.80 as follows: Is the trivial group the only group satisfying the normalizer condition and not containing a non-trivial abelian normal subgroup.

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    $\begingroup$ Problem 1.13 is a good example. As @YCor says above, questions about characterizations of the trivial group are not problems about the trivial group. For instance, one could phrase a famous question in geometric group theory as "Is the trivial group the only hyperbolic group without a non-trivial finite quotient?" Similarly, Problems 1.12 and 2.80 are not problems about the trivial group. $\endgroup$
    – HJRW
    Commented Mar 1, 2017 at 20:19
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In Group trisections and 4-smooth manifolds, the authors proved that the smooth 4-dimensional Poincaré conjecture is equivalent to the following (purely group theoretical) statement about the trivial group: Every $(3k, k)$–trisection of the trivial group is stably equivalent to the trivial trisection of the trivial group.

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