Automorphism Groups: Finite Cyclic G of Order n

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In summary, Aut(G) is the set of automorphisms of a finite cyclic group of order n that send a generator to a power k < n where (k,n) = 1. The group law is still unclear for Aut(Aut(G)). It is not necessarily cyclic and a better understanding of its structure is needed.
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Homework Statement



If G is a finite cyclic group of order n, what is the group Aut(G)? Aut(Aut(G))?

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The Attempt at a Solution



Aut(G) is given by the automorphisms that send a generator to a power k < n where (k,n) = 1 with order p(n) where p is Euler's function.

I'm having trouble visualizing or describing Aut(Aut(G)) as automorphisms of automorphisms. Is Aut(G) isomorphic to a cyclic group of order p(n)?
 
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  • #2
It is not going to be cyclic in general. You need to try to get a better view of Aut(G).

You've already identified it as a set as

{ k : 1<=k <n and (k,n)=1}

What is the group law?
 

FAQ: Automorphism Groups: Finite Cyclic G of Order n

What is an automorphism group?

An automorphism group is a mathematical concept that describes a set of transformations on a mathematical structure that preserve its properties. In other words, it is a group of symmetries that leave the structure unchanged.

What is a finite cyclic group of order n?

A finite cyclic group of order n is a group that consists of n elements and follows a cyclic pattern, meaning that there is a specific element that, when multiplied by itself, eventually cycles back to the identity element. An example of this is the group of rotations in a square, where the element of rotating by 90 degrees is cyclic with order 4.

How do automorphism groups relate to finite cyclic groups?

Automorphism groups can be used to describe the symmetries of a finite cyclic group. Each element in the automorphism group corresponds to a specific transformation that preserves the structure of the finite cyclic group. The automorphism group can also be used to determine the order of the cyclic group.

What are the properties of automorphism groups of finite cyclic groups?

Some properties of automorphism groups of finite cyclic groups include:

  • The order of an automorphism group is always equal to the order of the finite cyclic group.
  • The automorphism group is always cyclic.
  • For cyclic groups of prime order, the automorphism group is isomorphic to the group itself.
  • The automorphism group is abelian, meaning that its elements commute with each other.

How are automorphism groups of finite cyclic groups used in real-world applications?

Automorphism groups of finite cyclic groups are used in various areas of mathematics, such as group theory, number theory, and abstract algebra. They also have applications in computer science, particularly in cryptography and coding theory. Additionally, the concept of automorphism groups has been used in physics to describe symmetries in physical systems.

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