jud

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These are questions asked by jud

I can get a group like this:

g := SmallGroup(48, 8)

But I want to get the PermutationGroup form like PermutationGroup({[[...]], [[...], [...]]}). Can we change it into this form?

As we know, since the order about the element Perm([[1, 2], [3, 4, 5]]) of S5 is 6. Then  the C6 is a subgroup of S5, but why IsSubgroup(CyclicGroup(6), SymmetricGroup(5)) return false? Is it a bug?

Of course, we know the A5 of 60 order is an unsolvable group, but as the wiki here, There are also some solvable groups in the same 60 order. Similarly, although map(IsSolubleNumber, [60, 120, 168, 180]) will give false, there are some solvable groups in orders 60, 120, order 168, and order 180. But how to find these corresponding solvable polynomials by maple? I tried to generate them using random polynomials like this:

with(GroupTheory);
do
    do poly := randpoly(x, degree = rand(6 .. 8)()); until irreduc(poly);
    G := GaloisGroup(poly, x);
until IsSoluble(G) and is(GroupOrder(G) in {60, 120, 168, 180});
poly;
galois(poly, x);

But I didn't get any result even after one night..

I'm a newbie to maple, I want to get a group that meeting the criteria with this code:

with(GroupTheory);
G := SymmetricGroup(6);

do

H := Subgroup({RandomElement(G),RandomElement(G)}, G);

C := Core(H, G);

until is(GroupOrder(C) <> 1) and is(GroupOrder(C) <> GroupOrder(H));
H;
C;

But it's seem I cannot construct subgroup. so I adjust it into:

with(GroupTheory);
G := SymmetricGroup(6);

do

H := Subgroup({[[rand(1 .. 6)]], [[rand(1 .. 6), rand(1 .. 6)], [rand(1 .. 6), rand(1 .. 6)]]}, G);

C := Core(H, G);

until is(GroupOrder(C) <> 1) and is(GroupOrder(C) <> GroupOrder(H));
H;
C;

However, it is clear that random numbers do not always satisfy the rules of permutation. So how can I use the loop to generate a random subgroup that satisfies the condition?

with(GroupTheory);
G := GaloisGroup(x^5 + 20*x + 32, x);

I just can get a result like:

GroupTheory:-GaloisGroup(x^5 + 20*x + 32, x)

But how do I know exactly what group it is? Actually I know the galois group is D5, But can we use maple to get this information?

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