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You can define the ring of integers modulo n and then its unit group:

sage: n = 10
sage: R = IntegerModRing(n)
sage: G = R.unit_group()
sage: G
Multiplicative Abelian group isomorphic to C4
sage: G.order()
4
sage: G.inject_variables()
Defining f
sage: f^2
f^2
sage: f.order()
4
sage: (f^2).order()
2

You can define use the ring unit group of integers modulo n and then its unit group:your ring:

sage: n = 10
sage: R Zn = IntegerModRing(n)
Zmod(n)
sage: G = R.unit_group()
Zn.unit_group()
sage: G
Multiplicative Abelian group isomorphic to C4
sage: G.order()
4
sage: G.inject_variables()
Defining f
sage: f^2
f^2
sage: f.order()
4
sage: (f^2).order()
2

You can use the unit group of your ring:

sage: n = 10
sage: Zn = Zmod(n)
sage: G = Zn.unit_group()
sage: G
Multiplicative Abelian group isomorphic to C4
sage: G.order()
4
sage: G.inject_variables()
Defining f
sage: f^2
f^2
sage: f.order()
4
sage: (f^2).order()
2

You can use the unit group of your ring:

sage: n = 10
sage: Zn = Zmod(n)
sage: G = Zn.unit_group()
sage: G
Multiplicative Abelian group isomorphic to C4
sage: G.order()
4
sage: G.inject_variables()
Defining f
sage: f^2
f^2
sage: f.order()
4
sage: (f^2).order()
2

This group G comes with and embedding into Zn:

sage: Zn(f)
7
sage: Zn(f^2)
9
sage: Zn(f^3)
3
sage: Zn(f^4)
1
sage: Zn(f^5)
7
7