Show that all automorphisms of a group G form a group under function composition.

Show that all automorphisms of a group G form a group under function composition. Then show that the inner automorphisms of G, defined by f : G--->G so that f(x) = (a^(-1))(x)(a), form a normal subgroup of the group of all automorphisms. For the first part, I can see that we need to show that f(g(x)) = g(f(x)) for x in G and use f(x)=x as the identity in the group, but I' not certain how to proceed to show all innG form a normal subgroup?

Direct product problem

Let a be the permutation (1 2 3) in A_4. What is the order of the element (3, 7, a) in the group U(10) direct product Z_42 direct product A_4.

Isomorphism Problem

Show that U(10) is isomorphic to Z_4 and write out the isomorphism explicitly. I know that U(10) and Z_4 are both cyclic, thus they are ismorphic but for writing out the isomorphism, I need assistance.

Group question

Does the set {irrational numbers} U {1} form a group under multiplication? Either show this or explain why it is not true.

Is U(12) cyclic?

What is the complete multiplication table for U(12)? What are the elements in the subgroup <5>. is the group U(12) cyclic?