Group Theory V.4 (Optional: Automorphism Group)

Recall that for a set X, we let S_X denote the set of all bijective functions from X to itself. Now if X had some additional structure, we prefer to look at those functions which preserve the structure. For example, X is the set of points on the icosahedron, we look at the set of its symmetries preserving the geometric structure and there are only 120 of them.

Bearing this in mind, we define:

Let G be a group; an automorphism of G is an isomorphism from G to itself. Let Aut(G) be the set of all automorphisms of G.

Clearly, the inverse of an automorphism is another one; we can compose two automorphisms to give another, and the identity map is also an automorphism. Hence:

Aut(G) is a group under composition.

Let’s look at some examples of non-trivial automorphisms:

1. The inverse map which takes g to g^-1 is an automorphism if and only if G is abelian.
2. If G = C_n is the cyclic group of order n, written as {0, 1, …, n-1}, then an automorphism takes 1 to some a which generates the group. Such an a satisfies 0 ≤ a < n, with (a, n) = 1. Hence, an automorphism is of the form φ(i) = ai for some a. Composition of automorphisms corresponds to multiplication of a‘s. Hence, Aut(G) is isomorphic to (Z/n)*.
3. For any group G and element g of G, conjugation by g is an automorphism of G. This is called an inner automorphism.
4. Consider G = Z × Z. An automorphism of G takes (1, 0), (0, 1) to (a, b), (c, d) for some integers a, b, c, d. Since composition of the automorphisms corresponds to matrix multiplication, we see that the matrix must be invertible. Hence, Aut(G) is the set of 2 × 2 matrices with integer entries and determinant ±1.

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The case of inner automorphisms is of particular interest: we saw earlier that composition of two conjugations is another conjugation, so:

The set of inner automorphisms forms a subgroup Inn(G) of Aut(G).

Furthermore, it’s a normal subgroup! Indeed, for if f : G → G is any automorphism, and φ_g(h) = ghg^-1 is an inner automorphism, then:

which is precisely conjugation by f(g). Hence, we define:

The outer automorphism group Out(G) := Aut(G)/Inn(G).

One can visualise the outer automorphism group as follows. Generally, inner automorphisms are cheap and easy to find – in fact, every element of the group provides one. Thus, the Out(G) group attempts to ignore the cheap automorphisms in order to grasp the essence of Aut(G).