# Torsion

• Dec 11th 2010, 01:45 PM
DanielThrice
Torsion
Consider the (additive) factor group Q/Z.
(a) Show that every coset of Z in Q contains exactly one representative q ∈ Q in the range0 ≤ q < 1.
(b) Show that every element of Q/Z has ﬁnite order, but there are elements of arbitrarily large order.
(c) Show that Q/Z is the torsion subgroup of R/Z.
(d) Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unityin C. (That is, U∗ = {z ∈ C|z^n = 1 for some n ∈ Z+}.)

I got a and b, but I can't figure out c or d, any help?
• Dec 11th 2010, 03:17 PM
tonio
Quote:

Originally Posted by DanielThrice
Consider the (additive) factor group Q/Z.
(a) Show that every coset of Z in Q contains exactly one representative q ∈ Q in the range0 ≤ q < 1.
(b) Show that every element of Q/Z has ﬁnite order, but there are elements of arbitrarily large order.
(c) Show that Q/Z is the torsion subgroup of R/Z.
(d) Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unityin C. (That is, U∗ = {z ∈ C|z^n = 1 for some n ∈ Z+}.)

I got a and b, but I can't figure out c or d, any help?

(c) We have that if $r+\mathbb{Z}\in \mathbb{R}/\mathbb{Z}\,,\,\,ord(r+\mathbb{Z}) = n$ , then $nr\in\mathbb{Z}\Longleftrightarrow r\in \mathbb{Q}$

(d) Check the map $\phi: Q\rightarrow U^*\,,\,\, \phi(q):=e^{2\pi iq}$

Tonio