A torsion-free group is a group whose only element of finite order is its
identity element. Thus 
for a positive integer 
implies 
. Equivalently, its group
torsion is trivial. The infinite cyclic group 
and all free groups are examples,
whereas a nontrivial finite group is not torsion-free
(Brown 1982).
Every finite stabilizer in a proper group action of a torsion-free group is trivial. It also enters the Kadison-Kaplansky conjecture for projections in reduced group C*-algebras.
Nnuga Mongo (2026) proposed an infinite torsion-free nonabelian simple group realized as the reduced norm-one group
of a degree-15 central division algebra over
, with faithful
representations in 
and 
. This is a candidate
answer to problem 16.78 of the Kourovka Notebook. Nnuga Mongo (2026) states that
the construction and proof were generated by GPT-6.1 Sol and had not been independently
verified by human mathematicians. The claimed simplicity is therefore unverified.
