1. Let G be a group and let a be in G. An element b in G is called a conjugate of a if there exists an element x in G such that b = xax^-1. Show that any conjugate of a has the same order as a.

2. Let G be an abelian group and let x,y be in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) (order of xy) divides o(x)o(y).