Circle group

From Exampleproblems

In mathematics, the circle group, denoted by T (or in blackboard bold by $\mathbb T$ ), is the multiplicative group of all complex numbers with absolute value 1. The name comes from the fact that these numbers lie on the unit circle in the complex plane.

$\mathbb T = \{ z \in \mathbb C : |z| = 1 \}.$

The circle group forms a subgroup of C^×, the multiplicative group of all nonzero complex numbers. Since C^× is abelian, it follows that T is as well.

Isomorphisms

The circle group shows up in a huge variety of forms in mathematics. We list some of the more common forms here. Specifically, we show that

$\mathbb T \cong \mbox{U}(1) \cong \mbox{SO}(2) \cong \mathbb R/\mathbb Z.$

The set of all 1×1 unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to U(1), the first unitary group.

The exponential function gives rise to a group homomorphism exp : R → T from the additive real numbers R to the circle group T via the map

$\theta \mapsto e^{i\theta} = \cos\theta + i\sin\theta.$

The last equality is Euler's formula. The real number $θ$ corresponds to the angle on the unit circle as measured from the positive x-axis. That this map is a homomorphism follows from the fact the multiplication of unit complex numbers corresponds to addition of angles:

$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}.$

This exponential map is clearly a surjective function from R to T. It is not, however, injective. The kernel of this map is the set of all integer multiples of $2π$ . By the first isomorphism theorem we then have that

$\mathbb T \cong \mathbb R/2\pi\mathbb Z.$

After rescaling we can also say that T is isomorphic to R/Z.

If complex numbers are realized as 2×2 real matrices (see complex number), the unit complex numbers correspond to 2×2 orthogonal matrices with unit determinant. Specifically, we have

$e^{i\theta} \Leftrightarrow \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}.$

The circle group is therefore isomorphic to the special orthogonal group SO(2). This has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex plane, and every such rotation is of this form.

Lie group structure

The circle group is a compact Lie group, and any compact Lie group G of dimension ≥ 1 has a subgroup isomorphic to it. That means that, thinking in terms of symmetry, a compact symmetry group acting continuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen for example at rotational invariance, and spontaneous symmetry breaking.

The circle group has many subgroups, but its only closed subgroups consist of roots of unity: there is one such, that is cyclic of order n, for each integer n ≥ 1.

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Categories: Group theory | Topological groups | Lie groups