# Condition for Linear Operation on Complex Numbers to be of Finite Order

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## Theorem

Let $A$ be the operation on the complex numbers $\C$ defined as:

- $\map A x = \alpha x + \beta$

Then $A$ is of finite order greater than $1$ if and only if $\alpha$ is a root of unity other than $1$.

## Proof

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: Examples: $(3)$