[GAP Forum] Defining a group structure on a set

[Thomas Breuer]

Dear GAP Forum,

Timm von Puttkamer wrote:

> I have given a certain list of elements in GAP
> and I would like to define a group structure on this set of elements
> (so that I can use GAP to compute e.g. the center).
> As far as I see this should be possible using domains in GAP.
> However, I was not able to find out how to actually define
> the neutral element, inversion and multiplication.

I understand this question as follows.

You are given some objects which can be interpreted
as elements of a group or, more generally,
of a multiplicative structure which admits multiplying two objects,
inverting an object, and computing the identity object from any such object.

If the given objects cannot be multiplied with GAP's infix operator '*'
or if this multiplication is not the one which one has in mind
then the recommended way in GAP is to wrap the given objects
into new GAP objects and to define the relevant operations for them,
as is described in Chapter ``Designing Arithmetic Operations''
of the GAP Reference Manual;
perhaps the lightweight variant that is described in the manual section
``ArithmeticElementCreator'' is a good starting point.

All the best,
Thomas