# Finding non-trivial automorphisms of large Abelian groups

• jackmell
In summary, finding non-trivial automorphisms in large Abelian groups is significant for understanding the group's structure and properties, as well as for practical applications like cryptography. An automorphism in the context of Abelian groups is a bijective function that preserves the group operation. There is no specific method for finding non-trivial automorphisms, as it often involves a combination of reasoning, computation, and trial and error. Not all large Abelian groups will have non-trivial automorphisms, as it depends on the group's specific structure and properties. The size of the group can greatly affect the difficulty of finding non-trivial automorphisms, as larger groups have more complex structures and potential automorphisms.
jackmell
I feel there should be a way to find non-trivial (other than ##\operatorname{inn} G##) automorphisms of these groups other than by trial-and-error computation. Take for example ##\operatorname{aut}\mathbb{Z}_{100!}^*## . Are these just computationally inaccessable to us forever?

How about a simpler task: Now ##1_{G}(\mathbb{Z}_{100!}^*)=\{1,a_2,a_3,\cdots,n-1\}##. What is the least (even) number of substitutions I would have to make to obtain another automorphism? And can I figure that out other than by brute-force computation or is even brute-force searching computationally feasible for this group?

Or another:

Find a generator mapping ##\big\{a_1,a_2,\cdots,a_k\big\} \to \big\{b_1,b_2,\cdots,b_k\big\} ## that leads to a non-trivial automorphism other than by trial-and-error computation of residues.

Are there ways of doing these things for Abelian groups?

Ok thanks,
Jack

Last edited:

Dear Jack,

Thank you for bringing up this important question. I completely agree that finding non-trivial automorphisms of groups can be a difficult and time-consuming task. However, I do not believe that it is impossible to find such automorphisms without trial-and-error computation.

In fact, there are several techniques and algorithms that can be used to find non-trivial automorphisms of groups. One example is the use of generators and relations, which can help identify automorphisms based on the structure of the group. Another approach is to use character theory, which can provide information about the group's automorphisms through its irreducible representations.

Regarding your specific examples, I believe that it is possible to find non-trivial automorphisms of ##\operatorname{aut}\mathbb{Z}_{100!}^*## without brute-force computation. It may require some mathematical knowledge and creativity, but it is certainly feasible. Similarly, finding a generator mapping for a non-trivial automorphism can also be approached using the techniques mentioned above.

In conclusion, I do not think that these tasks are computationally inaccessible to us forever. With the right approach and tools, it is possible to find non-trivial automorphisms of groups without resorting to trial-and-error computation. I hope this helps alleviate your concerns and encourages you to continue exploring the fascinating world of group automorphisms.

## 1. What is the significance of finding non-trivial automorphisms in large Abelian groups?

Finding non-trivial automorphisms in large Abelian groups is important because it allows us to better understand the structure and properties of these groups. It also has practical applications in fields such as cryptography, where automorphisms are used to create secure encryption algorithms.

## 2. How do you define an automorphism in the context of Abelian groups?

An automorphism of an Abelian group is a bijective function that preserves the group operation. In other words, it is a mapping of the group elements onto themselves that does not change the group structure.

## 3. Is there a specific method or algorithm for finding non-trivial automorphisms in large Abelian groups?

There is no one specific method or algorithm for finding non-trivial automorphisms in large Abelian groups. It often involves a combination of theoretical reasoning, computational techniques, and trial and error.

## 4. Can non-trivial automorphisms be found in all large Abelian groups?

No, it is not guaranteed that all large Abelian groups will have non-trivial automorphisms. In fact, the existence of non-trivial automorphisms is dependent on the specific structure and properties of the group.

## 5. How does the size of the Abelian group affect the difficulty of finding non-trivial automorphisms?

The size of the Abelian group can greatly affect the difficulty of finding non-trivial automorphisms. Generally, larger groups will have more complex structures and more potential automorphisms, making it more challenging to identify them.

• Linear and Abstract Algebra
Replies
4
Views
2K
• Linear and Abstract Algebra
Replies
14
Views
3K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
11
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
2K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Linear and Abstract Algebra
Replies
12
Views
4K