Find the torsion subgroup of Z x Zn.?

Question

Z -> set of integers. Zn -> Z sub n.

Answer 1

The torsion subgroup is the set of all elements with finite order. Consider the element (a, b). The k'th "power" of this element under the group operation of addition will be (ka, kb modulo n). For (a, b) to be finite there must be some k > 0 for which this is the identity element (0, 0), i.e. ka = 0, kb = 0 (modulo n). Since k > 0, ka = 0 implies a = 0. Then we need b such that for some k, kb is a multiple of n. Clearly for any b, k = n will do. So the torsion subgroup is {0} x Zn.