1) Let G = <x> be a cyclic group of order n. Show that x^m is a generator of G if and only if (m,n) = 1. Thus the number of generators of a cyclic group of order n is the number of integers m in the set (0,1,...,n-1) such that (m,n) = 1. This is called Euler's function and plays a prominent role in number theory.
2) Let mZ and nZ be subgroups of (Z, +). What condition on m and n is equivalent to mZ is a subset of nZ? What condition on m and n is equivalent to mZ union nZ being a subgroup of (Z,+)?
(I don't really even get what this is asking...)
3) Prove that the intersection of two subgroups of a group G is itself a subgroup of G.
4) Give an example of a group G and a subset H of G such that H is closed under multiplication but H is not a subgroup of G.
5) a) Show that it is impossible for a group G to be the union of two proper subgroups.
b) Give an example of a group that is the union of three proper subgroups.