Automorphism

An automorphism is a mathematical term for a self-isomorphism in some category. More concretely, an automorphism of a group G is a group homomorphism G-->G which is an isomorphism. Likewise, an automorphism of a ring R is a bijective ring homomorphism from R to itself.

The collection of all automorphisms of a given object (a ring, group, vector space, metric space, topological space, graph,...) form a group with binary operation given by composition of functions, and the neutral element being the identity function that fixes every element of the object. This group can be thought of as a symmetry group for the object.

To illustrate the concept, we consider an example that is important in Galois theory. If L is a field extension of K. a function f:L-->L is a K-automorphism iff

1. f is a ring homomorphism.
2. f is a bijection
3. f(k)=k for all k in K

Here are some examples.

• Well first of all we always have the identity 1_L:L-->L defined by 1_L(a)=a for all a in L.
• C is a field extension of R. We have seen one R-automorphisms of C already, the identity, but there is another one. Complex conjugation gives a map bar:C-->C defined by bar(a+ib)=a-ib (for real numbers a,b). It is not difficult to check that this is indeed an R-automorphism of C.

  The question becomes are there any others? The answer is no. To see this suppose that f is some R-automorphism of C. Think about f(i). It is a complex number, what can it be? Well i²+1=0, and so f(i²+1)=f(0)=0. (The last equality is because f is a ring homomorphism.) But using that f is a ring homomorphism twice more we get that f(i²+1)=f(i)²+f(1). Finally, one more application tells us that f(1)=1. Putting it altogether we have f(i)²+1=0. In other words f(i) is a square root of -1 and so that tells us that f(i) is either i or -i.

  But once we know f(i) we know f. This is because a typical complex number looks like a+bi, for real numbers a,b. Now f(a+bi)=f(a)+f(b)f(i) (since f is a homomorphism) and since f fixes real numbers, f(a+bi)=a+bf(i).

  It follows that f is either the identity or complex conjugation.

Au`to*mor"phism (?), n.

Automorphic characterization.

H. Spenser.

 

© Webster 1913.