Modulo Multiplication is Commutative
Jump to navigation
Jump to search
Theorem
Multiplication modulo $m$ is commutative:
- $\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$
Proof
| \(\ds \eqclass x m \times_m \eqclass y m\) | \(=\) | \(\ds \eqclass {x y} m\) | Definition of Modulo Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {y x} m\) | Integer Multiplication is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass y m \times_m \eqclass x m\) | Definition of Modulo Multiplication |
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.7$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19.1$: Properties of $\Z_m$ as an algebraic system