lifting elements of a subgroup to the parent group

asked 2021-10-31 20:38:02 +0100

Max Alekseyev gravatar image

The following code defines a subgroup G of the unit group U of $\mathbb{Z}_{19}$:

Zn = Zmod(19)
U = Zn.unit_group()
u = U.gens()[0]
G = U.subgroup([u^3])
L = list(G)

and prints:

U: Multiplicative Abelian group isomorphic to C18
G: Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {f^3}
L: [1, f1, f1^2, f0, f0*f1, f0*f1^2]

How can I lift elements of G to U and then to Zn? For example, running U(L[0]) gives:

TypeError: 'AbelianGroup_subgroup_with_category.element_class' object is not iterable
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Do you imply that this issue is specific to just unit groups of $\mathbb Z_n$? Does lifting naturally works for other groups/subgroups?

Max Alekseyev gravatar imageMax Alekseyev ( 2021-11-13 20:57:57 +0100 )edit