The notation for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, (the direct product of with itself times) is geometrically an -torus.
One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° or or are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is 150° + 270° = 420°, but when thinking in terms of the circle group, we may "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360°, which gives 420° ≡ 60° (mod 360°).
Another description is in terms of ordinary (real) addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation: 360° or ), i.e. the real numbers modulo the integers: . This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out 0.4166... + 0.75, the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just with some preference to 0.166..., because .
The circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on , 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 (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 one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every -dimensional compact, connected, abelian Lie group is isomorphic to .
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 , the first unitary group.
The last equality is Euler's formula or the complex exponential. The real number θ corresponds to the angle (in radians) on the unit circle as measured counterclockwise from the positive x axis. That this map is a homomorphism follows from the fact that the multiplication of unit complex numbers corresponds to addition of angles:
Every compact Lie group of dimension > 0 has a subgroup isomorphic to the circle group. This 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 , the -th roots of unity form a cyclic group of order , which is unique up to isomorphism.
The representations of the circle group are easy to describe. It follows from Schur's lemma that the irreduciblecomplex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation
must take values in . Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself.
For each integer we can define a representation of the circle group by . These representations are all inequivalent. The representation is conjugate to :
The number of copies of must be (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of copies of is isomorphic to , as is a vector space of dimension over . Thus
can be proved in the same way, since is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of .