[GAP Forum] Does there exist an isomorphism such that...

[Alexander Hulpke]

Dear Forum,

On Oct 17, 2013, at 10/17/13 9:42, Le Van Luyen <lvluyen at gmail.com> wrote:

> Let G be a group with the order of Aut(G) very big.
> 
> Example:
> gap>G:=SmallGroup(256,56092);;
> gap> AutomorphismGroup(G);
> <group of size 5348063769211699200 with 2 generators>
> 
> We suppose that s,t: G->G are group homomorphisms.
> In GAP, Could you tell me a way to know whether there exists a group
> isomorphism f:G->G such that fs=tf?

I would try to first find, by having the automorphism group act on subgroups, an automorphism e, such that ker(s)=ker(t)^e and that Image(s)=Image(t)^e. After that we can assume WLOG that s and t have same kernel and image, and Aut(G) is reduced to the stabilizer A of these two subgroups.
Now you can look at these maps as isomorphism between G/ker(s) and Image(s). Go into the direct product of G/ker(s) with Image(s). For generators g1,g2,... of G/ker(s) you can represent s by the list of pairs (g1,g1^s),(g2,g2^s),... and t ditto.

Under the induced action of A on this direct product you now want to map the one list of pairs to the other.

How to do this best in practice might depend on the orders of kernel and image, feel free to send me an example for s and t and I'll have a look.

Best,

   Alexander Hulpke




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