Show that the set AutG of all automorphisms of a group G is

Show that the set Aut(G) of all automorphisms of a group (G,*) is a group under function composition and that the set Inn(G) of all inner automorphisms of a group (G,*) is a normal subgroup of Aut(G). (The factor group Aut(G)/Inn(G) is called the group of outer automorphisms of G, which is a misnomer as the elements of the factor group are not outer automorphisms nor do the outer automorphisms of G themselves form a group.) Please show all work. Thank You!

Solution

solution:

Let Aut(G) be the set of all automorphisms f: G --> G. In order to show that this is a group under the operation of composition, we must verify:

(1) Is the set is closed under composition? Yes! If you are given isomorphisms f, ?: G --> G, then it is not too tough to show that ?°f and f°? are isomorphisms. I can expand on this in more detail if you like, but you have probably seen a proof before that a composition of bijective functions is bijective. If a and b are elements of the group, ?°f(ab) = ?(f(ab)) = ?(f(a)f(b)), because f is an isomorphism. Since ? is also an isomorphism, ?(f(a)f(b)) = ?°f(a)?°f(b), so the composition ?°f preserves products. Thus, ?°f is an isomorphism if ? and f are.

(2) Is the set associative? Yes! All you need to do is show that, for any three isomorphisms f, ? and ?, f°(?°?) = (f°?)°?. To do that, just show that for each x in G, f°(?°?)(x) = (f°?)°?(x) = f(?(?(x))). It\'s just pushing around definitions.

(3) Does the set contain an identity element? Yes! Let the identity automorphism e: G --> G be the map e(x) = x. Clearly, e°f = f°e = f.

(4) Does each element of the set have an inverse under °? Yes! Since each isomorphism f: G --> G is bijective, there is a well-defined inverse map f^(-1): G --> G. You may have already seen a proof that the inverse of an isomorphism is an isomorphism. If not, it isn\'t too difficult to prove: I\'ll leave it to you, but I can expand on it if you need me to. Further, the composition f^(-1) ° f = f ° f^(-1) = e.

Since Aut(G) satisfies all the group axioms, it forms a group under °, as needed.