# Definition:Multiplicative Group of Complex Roots of Unity

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## Definition

Let $n \in \Z$ be an integer such that $n > 0$.

Let $U_n := \set {z \in \C: z^n = 1}$ denote the set of complex $n$th roots of unity.

Let $\struct {U_n, \times}$ be the algebraic structure formed by $U_n$ under complex multiplication.

Then $\struct {U_n, \times}$ is the **multiplicative group of complex $n$th roots of unity**.

## Also denoted as

Some sources denote this group as $K_n$.

## Also see

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Example $4.8$