[GAP Forum] Use GAP to Compute Hom(G1, G2)?

Jeffrey Rolland rollandj at uwm.edu
Sat Feb 9 01:37:07 GMT 2008

Alexander Hulpke wrote:
> Dear GAP Forum,
> Jeffrey Rolland asked:
>> I am trying to compute the set of all homomorphisms from a group G1 
>> [which is the semi-direct product of the integeres Z with the binary 
>> icosahedral group P (also known as SL(2,5) and the Poincare group)] to 
>> the group P (the Poincare group again) - Hom(G1, P). This sort of 
>> problem seems right up GAP's alley.
>> I have a presentatiion for G1: <z, s, t| s^3(st)^(-2), t^5(st^(-2), 
>> zs(s^2ts^2t^3z)^(-1), zt(s^5ts^2tz)^(-1)>.
> The easiest seems to be to find all quotients of G1 that are isomorphic 
> to a subgroup of SL(2,5). (There is some redundancy in this and for 
> bigger cases other methods would be better. However in this case 
> everything else is far more work for the user.)


> Careful: This group has no quotient isomorphic to A_5 and thus cannot 
> have SL(2,5) as quotient. So its probably not the group you want.


> Best,
>    Alexander Hulpke


Prof. Hulpke,

Oops! You are right. I actually want all homomorhphisms to Out(P) = Z_2. 
I know you can just send z from the Z in G1 to 1 in Z_2 and the s and t 
from the P in G1 to 0 in Z_2, but I needed to know if there were any 
other homs. Sorry, it's been a long time since I looked at this, and I 
forgot what I needed. I put this on the back burner until I realized GAP 
may be able to do this.

At any rate, your post should give me what I need. Thanks.

Jeffrey Rolland
<rollandj at uwm.edu>

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