# Finding non-trivial automorphisms of large Abelian groups

1. May 30, 2015

### 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: May 30, 2015
2. Jun 4, 2015