Cyclic Automorphism Group

[StudentR]

I came across a section of my notes that claimed the automorphism group of the cyclec groups C_2p where p=5,7,11 is cyclic,
that is Aut(C_2p) is cyclic for p = 5,7,11.
I wasn't able to see why this is so.
Is it just a fact or is there some sort of proof of the above...?
Thanks

 

[matt grime]

What is the automorphism group of the cyclic group? It sends a generator to a primitive power of itself. I.e. in C_m with generated g, any aut sends g to g^m for some m with (m,n)=1. You can work it out from there.

 

[tacman]

The statement that the automorphism group of cyclic groups C_2p for p=5,7,11 is cyclic is indeed a fact, and it can be proven using group theory and number theory concepts.

First, let's define what an automorphism group is. An automorphism of a group G is an isomorphism from G to itself. The set of all automorphisms of G forms a group under composition, called the automorphism group of G, denoted by Aut(G).

Now, let's consider the cyclic group C_2p, where p is a prime number. This group has p elements, and it is generated by a single element, say g. This means that every element of C_2p can be written as g^k, where k is an integer between 0 and p-1.

Now, an automorphism of C_2p is a map from C_2p to itself that preserves the group structure. Since C_2p is generated by g, any automorphism of C_2p must map g to another element of C_2p, say g^m, where m is also an integer between 0 and p-1. This means that there are p possible automorphisms of C_2p, given by the mapping g^k to g^(km) for k = 0,1,...,p-1.

To show that Aut(C_2p) is cyclic, we need to find a generator for this group. We can do this by considering the order of each element in Aut(C_2p). Since Aut(C_2p) has p elements, by Lagrange's theorem, the order of each element must divide p. This means that the possible orders of elements in Aut(C_2p) are 1 or p.

Now, let's consider the map f: C_2p -> C_2p given by f(g^k) = g^(k+1) for k = 0,1,...,p-1. It is easy to see that f is an automorphism of C_2p, and its order is p. This means that f is a generator of Aut(C_2p), and hence, Aut(C_2p) is cyclic.

In summary, the fact that Aut(C_2p) is cyclic for p = 5,7,11 is a result of the structure of cyclic groups and