The Unit Group

UnitGroup(R) : RngIntRes -> GrpAb, Map
Given R=Z/mZ, construct the unit group of R as an abelian group. This returns an (additive) abelian group A of order φ(m), together with a map from A to R.
IsPrimitive(n) : RngIntResElt -> BoolElt
Returns true if the element n∈Z/mZ is primitive, that is, if it generates the multiplicative group of Z/mZ, false otherwise.
PrimitiveElement(R) : RngIntRes -> RngIntResElt
PrimitiveRoot(R) : RngIntRes -> RngIntResElt
Given R = Z/mZ, this function returns a generator for the group of units of R if this group is cyclic, and returns 0 otherwise. Thus a valid generator is only returned if m = 2, 4, pt or 2pt, with p an odd prime and t≥1.
Order(a) : RngIntResElt -> RngIntElt
Given an element a belonging to Z/mZ, return the multiplicative order k ≥1 of a if a is in the unit group (Z/mZ)^*, and zero if a is not a unit.
Normalize(x) : RngIntRes -> RngIntResElt, RngIntResElt
Normalise(x) : RngIntRes -> RngIntResElt, RngIntResElt
Given an element x∈R = Z/mZ, this function returns the unique canonical associate y∈R of x and a unit u∈R such that u.x = y. The canonical associate of x is the GCD of x and m, considered as natural integers (unless x is 0, in which case it is 0).

Example RngIntRes_unit-group (H20E5)

We determine the unit group of the ring with modulus 735 and then verify its order by comparing it with φ(m). 
> m := 735;
> R := ResidueClassRing(m);
Residue class ring of integers modulo 735
> U, psi := UnitGroup(R);
> U;
Abelian Group isomorphic to Z/2 + Z/2 + Z/84
Defined on 3 generators
Relations:
    2*U.1 = 0
    4*U.2 = 0
    42*U.3 = 0
> #U;
336
> EulerPhi(735);
336

So the order of U is equal to φ(m) as it should be. Finally, we look for three elements of R that generate the unit group. 

> gens := [ psi(U.i) : i in [1..3] ]; gens;
> [ Order(x) : x in gens ];
[ 2, 4, 42 ]

Example RngIntRes_cyclic-unit-group (H20E6)

We construct a residue class ring R = Z/mZ having cyclic unit group. By a theorem of Gauss, the ring R has cyclic unit group precisely when n = 4, n = pe, or n = 2pe, and p is an odd prime. 
> R := IntegerRing(50);
> U, psi := UnitGroup(R);
Abelian Group isomorphic to Z/20
Defined on 1 generator
Relations:
    20*U.1 = 0
> w := PrimitiveElement(R);
> w;
3
> Order(w);
20

We verify that the powers of w are precisely the elements of the unit group U. 

> powers := { w^i : i in [0..19] };
> powers;
{ 29, 1, 31, 3, 33, 7, 37, 9, 39, 11, 41, 13, 43, 17, 47, 19, 49, 21, 23, 27 }
> powers eq { psi(u) : u in U };
true
V2.28, 13 July 2023