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A group in which the identity is the only element of finite order.
A group is torsion-free if
Equivalently, every nonidentity element has infinite order. The additive groups and , and every free group, are torsion-free; a nontrivial finite group is not.
This condition is inherited by subgroups. It is weaker than being orderable, although an orderable group is necessarily torsion-free.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a set. A free group on is a group together with an injection such that the following universal property holds:
For every group and every function , there exists a unique group homomorphism with .