Computer algebra systems are generally divided [1] into those that treat groups, or other mathematical structures, such as GAP, Cayley, and Magma, and those that treat polynomials and objects built from polynomials, such as Macsyma and its derivatives, Reduce, Maple, Axiom, and Mathematica.

This paper describes the GroupTheory package shipped in Maple 17, a replacement for the previous, and limited, group package. The author describes the ability to work with symbolic groups, the interface, and the visual capabilities, claiming that each has some capabilities that the existing group theory systems do not.

The ability to work with symbolic groups, such as D_{6n} where n is unspecified, is indeed different. In fact, it is not really computational group theory as such, but abstract reasoning from properties in databases. It is a pity that the example given in the paper, IsNilpotent (DihedralGroup (6*n)) assuming n :: posint, is shown as giving “true,” whereas the correct answer, which is actually what is returned by Maple, is “false.” This part of the project could do with being more fully explained, as it is an interesting question what one can do with this methodology.

The interface does contain some useful features building on Maple’s good mathematical interface: for example, DirectProduct (Monster (), Symm (n)) returns D₈ × D₈ rather than GAP’s “permutation group of size 64 with 6 generators.”

As far as visualization is concerned, the subgroup lattice capability is similar to that in GAP. It is again a pity that the example in the paper, DrawSubgroupLattice (g, highlight=LowerCentralSeries (g)) doesn’t work, and one needs DrawSubgroupLattice (g, highlight=[seq(h, h in LowerCentralSeries (g))]), which does indeed produce the diagram shown in the paper.

The Cayley table functionality does seem to be novel, but I am not sure how useful it would be in practice: if the group is large enough to be interesting, the Cayley table is probably unmanageable.

Will I abandon GAP for this package: No. Will I use it when I’m already computing in Maple: quite probably.