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.

nota problem about the trivial group. $\endgroup$4more comments