Integers under Subtraction do not form Group

Theorem

Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction.

Then $\struct {\Z, -}$ is not a group.

Proof

It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the group axioms.

First it is noted that Integer Subtraction is Closed.

Thus $\struct {\Z, -}$ fulfils Group Axiom $\text G 0$: Closure.

However, we then have Subtraction on Numbers is Not Associative.

So, for example:

$3 - \paren {2 - 1} = 2 \ne \paren {3 - 2} - 1 = 0$

Thus it has been demonstrated that $\struct {\Z, -}$ does not satisfy the group axioms.

Hence the result.

$\blacksquare$

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $1 \ \text{(a)}$