Modulo Multiplication is Commutative

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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.