[GAP Forum] central automorphism

[Stephen Linton]

I don’t think there is a library or package function that does this job.  
For small groups it should be easy enough to simply find all the central automorphisms from the
full automorphism group simply by using the definition and Filtered.

Here is an example of such a calculation performed interactively for D16:

gap> g := DihedralGroup(16);
<pc group of size 16 with 4 generators>
gap> a := AutomorphismGroup(g);
<group of size 32 with 5 generators>
gap> z := Centre(g);
Group([ f4 ])
gap> Filtered(a, x->ForAll(g, y->y^-1*Image(x,y) in z));  
[ [ f1*f2*f3*f4, f1*f3 ] -> [ f1*f2*f3*f4, f1*f3 ], [ f1*f2*f3*f4, f1*f3 ] -> [ f1*f2*f3, f1*f3*f4 ], 
  [ f1*f2*f3*f4, f1*f3 ] -> [ f1*f2*f3, f1*f3 ], [ f1*f2*f3*f4, f1*f3 ] -> [ f1*f2*f3*f4, f1*f3*f4 ] ]

The result is a list of four central automorphisms.

This approach would be horrendously inefficient for larger groups and I am sure there are much better ways.

You can find information about all the Gap Commands used here in the reference manual on the website, or using the
builtin help of GAP.

	Steve

> On 30 Apr 2015, at 18:51, abdulhakeem alayiwola <lovepgroups at gmail.com> wrote:
> 
> how do i find central automorphim of a group, say D8?
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