# Revision history [back]

Here is a slightly convoluted way to work around the current implementation defects.

sage: sage: list(prod((a^b for a, b in zip(H.gens(), h.list())), G.one()) for h in H)
[1, f^2, f^4]


Here is The current implementation has some defects

• elements of H have G as their parent
• but they display in terms of a generator of H

Find below a slightly convoluted way to work around around this.

Define G and H:

sage: n = 7
sage: Zn = Zmod(n)
sage: G = Zn.unit_group()
sage: f = G.gen()
sage: H = G.subgroup([f^2])


List the current implementation defects.

sage: sage: elements of H (this displays using a different f): sage: Hlist = list(H)
sage: Hlist
[1, f, f^2]
List the elements of H expressed in G: sage: HlistG = list(prod((a^b for a, b in zip(H.gens(), h.list())), G.one()) for h in H)
sage: HlistG
[1, f^2, f^4]
Get their values in Zn: sage: HlistZn = [h.value() for h in HlistG]
sage: HlistZn
[1, 2, 4]


 3 No.3 Revision updated 2021-06-26 21:22:26 +0200 The current implementation has some defects elements of H have G as their parent but they display in terms of a generator of H Find below a slightly convoluted way to work around this. Define G and H: sage: n = 7 sage: Zn = Zmod(n) sage: G = Zn.unit_group() sage: f = G.gen() sage: H = G.subgroup([f^2]) List the elements of H (this displays using a different f): sage: Hlist = list(H) sage: Hlist [1, f, f^2] List the elements of H expressed in G: sage: HlistG = list(prod((a^b for a, b in zip(H.gens(), h.list())), G.one()) for h in H) sage: HlistG [1, f^2, f^4] Get their values in Zn: sage: HlistZn = [h.value() for h in HlistG] sage: HlistZn [1, 2, 4] I opened a ticket to make this happen more naturally: Sage Trac ticket 32064: Improve unit groups of rings Z/nZ and their subgroups 4 No.4 Revision updated 2021-07-05 19:18:25 +0200 The current Going through the group of units This implementation has some defectssuffers from the defect that elements of H have G as their parent but they display in terms of a generator of H Find below Here is a slightly convoluted way to work around this. Define G and H: sage: n = 7 sage: Zn = Zmod(n) sage: G = Zn.unit_group() sage: f = G.gen() sage: H = G.subgroup([f^2]) List the elements of H (this displays using a different f): sage: Hlist = list(H) sage: Hlist [1, f, f^2] List the elements of H expressed in G: sage: HlistG = list(prod((a^b for a, b in zip(H.gens(), h.list())), G.one()) for h in H) sage: HlistG [1, f^2, f^4] Get their values in Zn: sage: HlistZn = [h.value() for h in HlistG] sage: HlistZn [1, 2, 4] I opened a ticket to make this happen more naturally: Sage Trac ticket 32064: Improve unit groups of rings Z/nZ and their subgroups Direct access to generating sets for subgroups The cyclic ring also has a method multiplicative_subgroups. That method lists generating tuples for its multiplicative subgroups: sage: n = 7 sage: Zn = Zmod(n) sage: Sub = Zn.multiplicative_subgroups() sage: Sub ((3,), (2,), (6,), ()) Sadly they do not give a hold on the subgroups as such. sage: H = Sub[1] sage: H (2,) sage: parent(H) <class 'tuple'> Neither do the generators for these subgroups have these subgroups as parents. Instead, they are simply elements of the initial cyclic ring. sage: h = H[0] sage: parent(h) Ring of integers modulo 7 


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