[GAP Forum] *some* outer automorphisms...

[Alexander Hulpke]

Dear Michael Hartley,

>
> is there any way to just find *some* outer automorphism(s) without
> finding the whole automorphism group?
>
> Ideally, I would want a method that would take a permutation group,  
> and
> find a subgroup of its automorphism group.
> The subgroup doesn't have to be the same each time. The method should
> be fast - faster than AutomorphismGroup(grp), of course.
> Also, if |Aut(grp)| >> |grp|, the method should have a "good  
> chance" of
> finding at least *some* outer automorphisms...

I think you will have to specify your problem a bit more: What  
tradeoff between runtime and found automorphisms do you want? What is  
a ``good chance''? Are you happy if one outer automorphism is found  
or do you want more? Are you content to only get outer automorphissms  
of a particular kind?

Also, what kinds of groups do you work with? Simple Groups? p-Groups?  
Solvable? (For example for simple groups, the question to find any  
automorphisms is basically of the same cost as finding all.)

The only straightforward method that come to mind for permutation  
groups is to calculate the normalizer in the symmetric group and  
consider the induced automorphisms -- however it really depends on  
the permutation representation on what automorphisms will be found.

Anything else would be too run the automorphism group algorithm(s)  
only partially, but what parts to run will depend on choices about  
tradeoff you will have to make.

Best wishes,

     Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke