Circle Group

Theorem

Let $\mathbb S$ be the set of all complex numbers of unit modulus:

$\mathbb S = \left\{{z \in \C: \left|{z}\right| = 1}\right\}$.

Then $\left({\mathbb S, \cdot}\right)$ is an infinite abelian group under the operation of complex multiplication.

This is called the circle group or the complex unit circle (group).

Proof

We note that $\mathbb S \ne \varnothing$ as the identity element $1 + 0 i \in \mathbb S$.

Since all $z \in \mathbb S$ have modulus $1$, they have, for some $\theta \in \left[{0 .. 2 \pi}\right)$, the polar form:

$z = \exp \left({i \theta}\right) = \cos \left({\theta}\right) + i \sin \left({\theta}\right)$

Conversely, if a complex number has such a polar form, it has modulus $1$.

Observe the following property of the exponential function:

$\forall a, b \in \C: \exp \left({a + b}\right) = \exp \left({a}\right) \exp \left({b}\right)$

We must show that if $x,y \in \mathbb S$ then $x\cdot y^{-1} \in \mathbb S$.

Let $x, y \in \mathbb S$ be arbitrary. Choose suitable $s, t \in \left[{0 .. 2 \pi}\right)$ such that:

$x = \exp \left({i s}\right)$

$y = \exp \left({i t}\right)$

We compute:

	\(\displaystyle \)	\(\displaystyle \exp \left({i t}\right) \exp \left({-i t}\right)\)	\(=\)	\(\displaystyle \exp \left({ i \left({t - t}\right) }\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \exp \left({0}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)

So $y^{-1} = \exp \left({-i t}\right)$. We note that this lies in $\mathbb S$.

Furthermore, we have:

	\(\displaystyle \)	\(\displaystyle xy\)	\(=\)	\(\displaystyle \exp \left({i s}\right) \exp(-2 \pi i t)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \exp \left({ i \left({s - t}\right) }\right)\)	\(\displaystyle \)

We conclude that $xy \in \mathbb S$.

By the Two-Step Subgroup Test, $\mathbb S$ is a subgroup of $\C$ under complex multiplication.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 7$: Exercise $7.2$
• Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.08$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 31$