Finding non-trivial automorphisms of large Abelian groups

[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

 

[jvicens]

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.