Groups are cancellative

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Statement

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.

Proof

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.