# Circle group

In mathematics, the **circle group**, denoted by or ^{1}, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the **unit complex numbers**[1]

Algebraic structure → Group theoryGroup theory |
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Lie groups |
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The circle group forms a subgroup of , the multiplicative group of all nonzero complex numbers. Since is abelian, it follows that is as well. The circle group is also the group of 1×1 complex-valued unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle of rotation by

This is the exponential map for the circle group.

The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.

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.

The circle group is isomorphic to the special orthogonal group .