# Integers under Addition form Abelian Group

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## Theorem

The set of integers under addition $\struct {\Z, +}$ forms an abelian group.

## Proof

From the definition of the integers, the algebraic structure $\struct {\Z, +}$ is an isomorphic copy of the inverse completion of $\struct {\N, +}$.

From Natural Numbers under Addition form Commutative Semigroup, $\struct {\N, +}$ is a commutative semigroup.

From Natural Number Addition is Cancellable all elements of $\struct {\N, +}$ are cancellable.

The result follows from Inverse Completion of Commutative Semigroup is Abelian Group.

Thus addition on $\Z$ is well-defined, closed, associative and commutative on $\Z$.

$\Box$

Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the formal definition of integers.

That is, $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

$\boxminus$ is the congruence relation defined on $\N \times \N$ by:

- $\tuple {x_1, y_1} \boxminus \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxminus$, as suggested.

### Identity is Zero

From Construction of Inverse Completion: Identity of Quotient Structure, the identity of $\struct {\Z, +}$ is $\eqclass {c, c} {}$ for any $c \in \N$:

\(\ds \forall a, b, c \in \N: \, \) | \(\ds \) | \(\) | \(\ds \eqclass {a, b} {} + \eqclass {c, c} {}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \eqclass {a, b} {}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \eqclass {c, c} {} + \eqclass {a, b} {}\) |

$\eqclass {c, c} {}$ is the equivalence class of pairs of elements $\N \times \N$ whose difference is zero.

Thus the identity of $\struct {\Z, +}$ is seen to be $0$.

Note that a perfectly good representative of $\eqclass {c, c} {}$ is $\eqclass {0, 0} {}$.

This usually keeps to a minimum the complexity of any arithmetic that is needed.

$\Box$

### Construction of Inverses

From Construction of Inverse Completion: Invertible Elements in Quotient Structure, we see that every element of $\struct {\Z, +}$ has an inverse.

We can see that:

\(\ds \forall a, b \in \N: \, \) | \(\ds \eqclass {a, b} {} + \eqclass {b, a} {}\) | \(=\) | \(\ds \eqclass {a + b, b + a} {}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \eqclass {a + b, a + b} {}\) | Natural Number Addition is Commutative |

The above construction is valid because $a$ and $b$ are both in $\N$ and hence cancellable.

From Construction of Inverse Completion: Identity of Quotient Structure, $\eqclass {a + b, a + b} {}$ is a member of the equivalence class which is the identity of $\struct {\Z, +}$.

Thus the inverse of $\eqclass {a, b} {}$ is $\eqclass {b, a} {}$.

$\blacksquare$

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(ii)}$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.5$. Examples of groups: Example $80$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.1$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$: Example $1$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.01$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 32$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Mappings: Exercise $9$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(1)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(i)}$ - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Abelian group** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**group** - 1999: J.C. Rosales and P.A. García-Sánchez:
*Finitely Generated Commutative Monoids*... (previous) ... (next): Chapter $1$: Basic Definitions and Results - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Abelian group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**group** - 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$

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- 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$: Example $1$