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What construct for cyclic group?

asked 2020-07-29 14:16:41 -0500

What is the proper Sage construct for working with cyclic group $\mathbb{Z} / n \mathbb{Z}$ ? I cannot see any in group theory manual page. Yes, we have Integers(n), but it cannot be asked like a group about, say, subgroup().

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answered 2020-07-30 01:39:34 -0500

philipp7 gravatar image

You can use the method CyclicPermutationGroup(n). This will create a cyclic group of given order. Then you can apply all the usual group-theory methods, e.g.

sage: G = CyclicPermutationGroup(8)
sage: G.is_cyclic()
True
sage: genG = G.gen()
sage: genG 
(1,2,3,4,5,6,7,8)

One problem is, that the elements are now represented as permutations and not in the "usual way" as integers $0,...,n-1$. However, if you want the element in $G$ which corresponds to $i \in \mathbb{Z}_n$ you can use genG^i.

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Asked: 2020-07-29 14:16:41 -0500

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Last updated: Jul 30