Exercise 4.1. (1) Let : G → G' be an isomorphism between two groups G and G'. Suppose G is cyclic. Show that G' is also cyclic. (2) An automorphism of a group G is an isomorphism from G onto itself. Determine the number of automorphisms of the following groups (i) (Z,+) (ii) (Z15, +15).

Exercises 35. Prove that any two groups of order are isomorphic.

For a fixed group G, prove that the set of all automorphisms of G forms a group with respect to mapping composition.

32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .