Groups are cancellative

From Groupprops
Jump to: navigation, search


Verbal statement

Any group is a cancellative monoid: every element in it is cancellative.

Symbolic statement

Suppose G is a group with binary operation *, and a,b,c \in G are elements such that:

a * b = a * c

then b = c

In other words a is left cancellative. A similar proof shows that a 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.