Although the Abstract Algebra Add-on is a beautiful package for Mathematica, imo there is nothing that beats GAP, at least not for Group Theory. When I looked at Sage a few years ago, I found that said to have an interface with Mma but when I looked at it closer it turned out to be somewhat primitive ( but operational ). I know that work has been done at the Technical University of Eindhoven (OpenMath) on a platform for integrating / interfacing the major math tools, I even started to work on it. But that work got lost. Now to my question:
Is there, does anyone have, and are you willing to share, an interface between GAP and Mathematica?
One options is to use Sage as an intermediate. Sage can interface both with Mathematica and GAP.
I believe that Sage interfaces with both programs by:
a) running their console interfaces in the background using pexpect
b) knowing how to translate most of the GAP and Mathematica syntax into its native syntax.
Note that I’ve done this a couple of times before and it works ok – but it means you don’t get to use the Mathematica notebook interface….
A really useful tool would be to hook Mathematica up to expect (or pexpect) so that similar interfaces to console programs can be written for Mathematica. This is basically what the second quote in Sjoerd’s answer is suggesting.