Finite Cyclic Group has Euler Phi Generators
Jump to navigation
Jump to search
Theorem
Let $C_n$ be a (finite) cyclic group of order $n$.
Then $C_n$ has $\map \phi n$ generators, where $\map \phi n$ denotes the Euler $\phi$ function.
Proof
From List of Elements in Finite Cyclic Group, the elements of $G$ are:
- $\set {g^k: g \in G, 0 \le k < n}$
From Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order, $g^k$ generates $G$ if and only if $k \perp n$.
The result follows by definition of the Euler $\phi$ function.
$\blacksquare$
Sources
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts: Exercise $5 \ \text{(ii)}$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups: Exercise $1$