[GAP Forum] Contain a fixed subgroup

[Alexander Konovalov]

Dear Mr. Sorouhesh, dear GAP Forum,

On 11 Jul 2012, at 21:07, Mr. Sorouhesh wrote:

> Dear Friends,
> 
> Trough working on the following problem:
> 
> http://math.stackexchange.com/q/168180
> 
> you can see, the questioner defined a* new operation *on CyclicGroup(7) ,
> to find inverse elements and search some properties in the group.
> 
> This crossed my mind with this question if we can define a new well-defined
> operation on a certain set ,in GAP, to do for example what he did in the
> above question?

Of course - allowing to define new objects and operations is one of the 
central GAP design principles.

There is a function called ArithmeticElementCreator which offers a simple 
interface to create new arithmetic elements by providing functions that 
perform addition, multiplication etc. See `?ArithmeticElementCreator' to
see the documentation and example.

One can get further with an approach described in the Reference manual,
Chapter 82 "An Example – Designing Arithmetic Operations", while
ArithmeticElementCreator may be useful to get started though does not 
permit to exploit all potential features.

See also the Circle package <http://www.cs.st-andrews.ac.uk/~alexk/circle/>
which provides functionality to construct circle objects that will respect 
the circle multiplication r *s = r + s + rs, to create multiplicative 
structures, generated by such objects, and to compute adjoint semigroups 
and adjoint groups of finite rings. We were trying to make Chapter 2
"Implementing circle objects" useful for users who want to implement their
own objects and operations following our examples.

Best regards,
Alexander