Reduced Residue System under Multiplication forms Abelian Group/Corollary
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Corollary of Reduced Residue System under Multiplication forms Abelian Group
Let $p$ be a prime number.
Let $\Z_p$ be the set of integers modulo $p$.
Let $\struct {\Z'_p, \times}$ denote the multiplicative group of reduced residues modulo $p$.
Then $\struct {\Z'_p, \times}$ is an abelian group.
Proof
Suppose $p \in \Z$ be a prime number.
From the definition of reduced residue system modulo $p$, as $p$ is prime, $\Z'_p$ becomes:
- $\set {\eqclass 1 p, \eqclass 2 p, \ldots, \eqclass {p - 1} p}$
This is precisely $\Z_p \setminus \set {\eqclass 0 p}$ which is what we wanted to show.
The result follows from Reduced Residue System under Multiplication forms Abelian Group.
$\blacksquare$
Sources
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.11$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 34$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $7$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(3)$