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==Sample session==
=== [[Permutation group]] ===
{{sxhl|lang=gap-console|1=
gap> G:=SmallGroup(8,1); # Set G to be the 1st group (in GAP catalogue) of order 8.
Line 54 ⟶ 55:
(1,5,3,7,2,6,4,8), (1,6,3,8,2,5,4,7), (1,7,4,5,2,8,3,6), (1,8,4,6,2,7,3,5) ]
}}
=== [[Euclidean ring]] ===
{{sxhl|lang=gap-console|1=
gap> # test consistency of EuclideanDegree, EuclideanQuotient, EuclideanRemainder,
gap> # and QuotientRemainder for some ring and elements of it
gap> checkEuclideanRing :=
> function(R, colls...)
> local coll1, coll2, a, b, deg_b, deg_r, q, r, qr;
> if Length(colls) >= 1 then coll1:=colls[1];
> elif Size(R) <= 100 then coll1 := R;
> else coll1 := List([1..100],i->Random(R));
> fi;
> if Length(colls) >= 2 then coll2:=colls[2];
> elif Size(R) <= 100 then coll2 := R;
> else coll2 := List([1..100],i->Random(R));
> fi;
> for b in coll1 do
> if IsZero(b) then continue; fi;
> deg_b := EuclideanDegree(R, b);
> for a in coll2 do
> q := EuclideanQuotient(R, a, b); Assert(0, q in R);
> r := EuclideanRemainder(R, a, b); Assert(0, r in R);
> if a <> q*b + r then Error("a <> q*b + r for ", [R,a,b]); fi;
> deg_r := EuclideanDegree(R, r);
> if not IsZero(r) and deg_r >= deg_b then Error("Euclidean degree did not decrease for ",[R,a,b]); fi;
> qr := QuotientRemainder(R, a, b);
> if qr <> [q, r] then Error("QuotientRemainder inconsistent for ", [R,a,b]); fi;
> od;
> od;
> return true;
> end;;
gap> # rings in characteristic 0
gap> checkEuclideanRing(Integers,[-100..100],[-100..100]);
true
gap> checkEuclideanRing(Rationals);
true
gap> checkEuclideanRing(GaussianIntegers);
true
gap> checkEuclideanRing(GaussianRationals);
true
gap> # finite fields
gap> ForAll(Filtered([2..50], IsPrimePowerInt), q->checkEuclideanRing(GF(q)));
true
gap> # ZmodnZ
gap> ForAll([1..50], m -> checkEuclideanRing(Integers mod m));
true
gap> checkEuclideanRing(Integers mod ((2*3*5)^2));
true
gap> checkEuclideanRing(Integers mod ((2*3*5)^3));
true
gap> checkEuclideanRing(Integers mod ((2*3*5*7)^2));
true
gap> checkEuclideanRing(Integers mod ((2*3*5*7)^3));
true
}}<ref>https://pygments.org/docs/lexers/#pygments.lexers.algebra.GAPConsoleLexer</ref>
==See also==
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