In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane.
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. The circle group is also the group U(1) of 1×1 unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ 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 T for the circle group stems from the fact that Tn (the direct product of T with itself n times) is geometrically an n-torus. The circle group is then a 1-torus.
Read more about Circle Group: Elementary Introduction, Topological and Analytic Structure, Isomorphisms, Properties, Representations, Group Structure
Famous quotes containing the words circle and/or group:
“... in any war a victory means another war, and yet another, until some day inevitably the tides turn, and the victor is the vanquished, and the circle reverses itself, but remains nevertheless a circle.”
—Pearl S. Buck (1892–1973)
“Caprice, independence and rebellion, which are opposed to the social order, are essential to the good health of an ethnic group. We shall measure the good health of this group by the number of its delinquents. Nothing is more immobilizing than the spirit of deference.”
—Jean Dubuffet (1901–1985)