[GAP Forum] Multiplicative Group modulo n

[Alexander Hulpke]

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,
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