Non-trivial problems about the trivial group 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?
 A: 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. 
A: 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.
