For (a), you need to verify the group axioms for A(G).

The identity element of A(G) is the trivial automorphism sending each element to itself and the inverse of an automorphism is its inverse as a set map.

For (b) and (c), to show that is an automorphism, you need to show first that is well-defined. Then, you need to show that is homomorphism such that . To show injectivity of , you need to show that if , then x=y, where . To show surjectivity, it suffice to show that , where .

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This is some additional stuff.

Those maps in (b) is called aninner automorphismof G. If and , then , which shows . Additionally, outer automorphism can be defined as .