# Cancellation Laws/Proof 2

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## Theorem

Let $G$ be a group.

Let $a, b, c \in G$.

Then the following hold:

- Right cancellation law

- $b a = c a \implies b = c$

- Left cancellation law

- $a b = a c \implies b = c$

## Proof

From its definition, a group is a monoid, all of whose elements have inverses and thus are invertible.

From Invertible Element of Monoid is Cancellable, it follows that all its elements are therefore cancellable.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Theorem $7.1$ Corollary