Abstract Algebra - automorphism

[nebbish]

I have two problems I would like to discuss.

1.For any group G prove that the set of inner automorphisms J(G) is a normal subgroup of the set of automorphisms A(G).

Let A be an automorphism of G. Let $$T_{g}$$ be an inner automorphism, i.e.
$$xT_{g}=g^{-1}xg$$

The problem can be reduced to the question whether the following equality is true:
$$xAT_{g}=xT_{g}A$$

Then expanding using $$xT_{g}=g^{-1}xg$$ we have:
$$gxAg^{-1}=gxg^{-1}A$$

However I am having trouble proving this equality. Attempting to use the definition of normal more directly also did not work, i.e. showing that
$$AT_{g}A^{-1}$$ is in J(G).

2.Let G = {e, a, b, ab} where $$a^2=b^2=e$$ and ab = ba. Determine the set of automorphisms A(G).
This problem could be handled easily using brute force but I would like some way to narrow down the possible automorphisms. My attempts to solve the problem so far have resulted in way too many calculations.

 

[matt grime]

2. Really, way to many calculations? There are only 6 possible permutations of a,b,ab to consider.1. If you are still strugglng with this, what inner automorphism do you think you ought to get for AT_gA^{-1}? It is T_h for some h, try taking a reasonable guess as to what h ought to be.