TITLE: Non-trivial problems about the trivial group
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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.