Thread: 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?

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 , then

(d) Check the map

Tonio