automorphism group of a cyclic group
Choose a generator for . If , then for some integer (defined up to multiples of ); further, since generates , it is clear that uniquely determines . Write for this automorphism. Since is an automorphism, is also a generator, and thus and are relatively prime11 If they were not, say , then so that would not generate.. Clearly, then, every relatively prime to induces an automorphism. We can therefore define a surjective map
is also obviously injective, so all that remains is to show that it is a group homomorphism. But for every , we have
- 1 Dummit, D., Foote, R.M., Abstract Algebra, Third Edition, Wiley, 2004.
|Title||automorphism group of a cyclic group|
|Date of creation||2013-03-22 18:42:35|
|Last modified on||2013-03-22 18:42:35|
|Last modified by||rm50 (10146)|