Everything about Circle Group totally explained
In
mathematics, the
circle group, denoted by
T (or in
blackboard bold by
), is the multiplicative
group of all
complex numbers with absolute value 1, for example, the
unit circle in the complex plane.
» is equivalent to
.
Algebraic structure
In this section we'll 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
»
The isomorphism
»
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.
Further Information
Get more info on 'Circle Group'.
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