Groups are cancellative
Suppose is a group with binary operation , and are elements such that:
In other words is left cancellative. A similar proof shows that is right cancellative. In other words, given equations in terms of elements of the group, we can always cancel elements from the left and from the right.
The proof follows from a somewhat more general fact: in a monoid (a set with associative binary operation and having identity element), any invertible element is cancellative: it can be canceled from the left or the right of any equation.