[GAP Forum] Representing groups as linear groups

[Johannes Hahn]

Hi.

There is no unique faithful linear representation for a arbitrary finite 
group in general. (For arbitrary groups there is in general not a single 
faithful linear representation)
I guess the only canonical choice is the regular representation, i.e. 
the group of permutation matrices that corresponds to the permutation 
action of G on itself via left-translation (or right-translation if you 
choose to act from the right). Therefore your way should be the easiest 
one to implement. Caveat: In general this procedure does not give you a 
faithful representation of smallest possible degree, i.e. it might be 
possible to find a "better" embedding in the sense that the matrices 
have smaller dimension. I don't think that there is some good way to do 
that besides the brute-force-way: Look through all characters of the 
group and check which are faithful. Then find a representation for this 
character (which is itself a nontrivial problem).

Johannes Hahn.

Am 21.11.2011 14:36, schrieb Robert Heffernan:
> Hi,
>
> Given a group G in GAP (eg. a group from the Small Groups library) is
> there an easy way to get the group as a subgroup of a linear group?
>
> I guess I could run IsomorphismPermGroup(G) and then get permutation
> matrices for the generators.  Is there a better way to do it than
> this?  A browse through the documentation didn't turn up anything
> obvious.
>
> Thank you,
> Bob Heffernan
>
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum