One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore we adjust our answer by 360° which gives 420° − 360° = 60°.
Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed. To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0.784 + 0.925 + 0.446, the answer should be 2.155, but we throw away the leading 2, so the answer (in the circle group) is just 0.155.
The circle group is more than just an abstract algebraic group. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on C×, the circle group has the structure of a topological group. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of C× (itself regarded as a topological group).
One can say even more. The circle is a 1-dimensional real manifold and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a 1-dimensional Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to Tn.
The circle group shows up in a variety of forms in mathematics. We list some of the more common forms here. Specifically, we show that
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 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:
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
After rescaling we can also say that T is isomorphic to R/Z.
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.
Any compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group. 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 proper closed subgroups consist of roots of unity: For each integer n > 0, the nth roots of unity form a cyclic group of order n, which is unique up to isomorphism.
The representations of the circle group are easy to describe. It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation ρ : T → GL(1, C) ≅ C×, must take values in U(1)≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself. Every such homomorphism is of the form
These representations are all inequivalent. The representation φ-n is conjugate to φn,
In this section we will forget about the topological structure of the circle group and look only at its algebraic structure.
The circle group T is a divisible group. Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. Thus
can be proved in the same way, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T.