1. Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag.

i) Find all possible conjugates f the permutation

ii) Find the centralizer p in .

iii) Prove that for any element a in a group G, the centralizer is a subgroup of G.

2. Let G= , the non zero real numbers, with operation given by multiplication. Let H be , the positive real numbers, with operation given by multiplication. You may assume that G and H are groups. Let be the map given by .

i) Prove that is a group homomorphism.

ii) Compute ker and im , explaining your answer