[GAP Forum] Multiplicative Group modulo n
Alexander Hulpke
hulpke at math.colostate.edu
Wed Dec 15 18:14:30 GMT 2010
Dear GAP Forum,
On Dec 5, 2010, at 12/5/10 6:57, Johannes Wachs wrote:
> Is there a way to call up the multiplicative group of order n?
You could create Z/nZ in GAP as
Units(Integers mod 75);
(or whatever modulus you like).
>
> In particular I have written a program to output some set of positive
> integers. I want to test if they are a subgroup of the multiplicative group
> mod n. It would be great if I could use Subgroup(G, L);
>
> Alternatively, I think I could solve my problem by checking to see if my set
> is closed under multiplication mod n. Is there a way to implement IsGroup
> for some set mod n with multiplication?
If you create a group in GAP, the convention is that GAP will take the multiplicative closure of this generating set. Thus -- unless you wanted to compare elements -- `Subgroup' will not help for testing whether a set is a group.
Similarly `IsGroup' is declared in GAP as a category, i.e. a property that an object will have from creation, but that is not testable. (I.e. if you define a semigroup, even if it turns out to be a group, it will never fulfill `IsGroup'.)
In the case of your problem, I fear that fundamentally you will have to test closure (though the fact that it is a subset of an abelian group with known base will allow for some shortcuts).
Best,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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