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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?

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Solution Preview

First part: Let Aut(G) be the set of all automorphisms of a group G.
To show that Aut(G) is a group under function composition, we need to
verify the following three facts.
(1) id(x)=x is the identity of Aut(G). Because for any f in Aut(G) and
any x in G, we have id(f(x))=f(x)=f(id(x)). So ...

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Aut(G), the set of automorphisms of G, is also a group.

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If G is a group, then Aut(G), the set of automorphisms of G, is also a group.

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