23
votes
Accepted
How to solve this solvable 8th-degree algebraic equation by radicals?
Well, in honor of an old cartoon I'll say a miracle occurs. But can we get behind the curtain to see how the special effects are made?
If you take the square root of, let us say, $2358$ by the ...
- 37.5k
19
votes
Accepted
How is GAP generating all subgroups?
GAP stores precomputed lists of subgroups of (automorphism groups) of a number of simple subgroups, the rest is computed on the fly. However it uses far more group theory than only considering subsets....
- 17.6k
18
votes
Accepted
How to find all subgroups of a group in GAP
One should think in terms of conjugacy classes of subgroups:
...
- 6,952
18
votes
Accepted
How to check in GAP whether two groups are isomorphic
There are two aspects here: how to search such information in GAP, and how to actually check the isomorphism of two groups.
First, the OP was quite close to guessing the name of the function, since ...
- 6,952
17
votes
Accepted
Is $\mathrm{gnu}(2304)$ known?
There are indeed $112\,184+1\,953+15\,641 993 = 15\,756\,130$ groups of order 2304, computed using an algorithm developed by Bettina Eick and myself. As Alexander Konovalov already kindly pointed out, ...
- 1,812
17
votes
Accepted
Proof that the Rubik’s Cube group is 2-generated
To start, we will use the example from "Analyzing Rubik's Cube with GAP". It creates the group generated by the six generators, corresponding to the six faces of the cube:
...
- 6,952
11
votes
Accepted
List Conjugacy Classes in GAP?
Let's create some group as an example:
gap> G:=Group((1,9,6,7,5)(2,10,3,8,4), (1,10,7,8)(2,9,4,6));
Group([ (1,9,6,7,5)(2,10,3,8,4), (1,10,7,8)(2,9,4,6) ])
It ...
- 6,952
11
votes
Accepted
Groups of derangements: what is known about subgroups of a symmetric group $S_{n}$ that contain only derangements (plus the identity)?
Such a group is sometimes called semiregular (or free, or fixed point free, although the later two are more often reserved for group actions than permutation groups), see
https://en.wikipedia.org/wiki/...
- 6,572
10
votes
Accepted
Non-isomorphic groups with identical structure-description
StructureDescription will -- despite what was claimed in older implementations -- not identify groups up to isomorphism, but just indicate a decomposition. For ...
- 17.6k
10
votes
If I kill any element $g$ of a group $G$ and as a result $G/\langle\langle g\rangle\rangle$ is killed, is $G$ then cyclic?
The answer is "no". The issue is that the normal subgroup $\langle\langle g \rangle \rangle < G$ that is generated by $g$ can be much larger and more complicated than the ordinary subgroup $\langle ...
- 116k
10
votes
How do I create a permutation group from the Small Group Library?
The command IsomorphismPermGroup( G ) will construct an isomorphism from a given group $G$ to a subgroup of $S_{|G|}$. You can then work with the image of this ...
- 6,565
9
votes
Accepted
How do Gap generate the elements in permutation groups?
Alex's answer is very good, but I'd like to add a somewhat simplified explanation of the Schreier-Sims Algorithm, since @Lehs asked for it.
I want to emphasis that this is not the actual Schreier-...
- 2,386
9
votes
Accepted
Mysterious computation of a ring quotient in GAP
Yes, this is two (stupid) bugs I will fix in the next release. Thank you for reporting.
I will put a temporary patch (just read in the file) at
https://www.dropbox.com/s/aprqfottc4pb6ni/quotringfix.g?...
- 17.6k
9
votes
Accepted
How does GAP calculate irreps?
Assuming you have a finite group and you ask for the calculation of irreducible characters over $\mathbb C$, GAP will as default method use the Dixon-Schneider algorithm: [MR1075426,
Schneider, ...
- 17.6k
9
votes
representation of the dihedral group $D_3$
This is basically a two-stage process. First we compute (in the representation you have) the idempotents for the different irreducible representations. They are given by the formula
$$
e_\chi=\frac{1}{...
- 17.6k
9
votes
How to solve this solvable 8th-degree algebraic equation by radicals?
Just to explain why the attempt to calculate in GAP failed. The error message was produced by the Alnuth GAP package on which RadiRoot depends, and Alnuth in its turn requires PARI/GP. If everything ...
- 6,952
8
votes
Which resources are available to self-study GAP?
A book called "Computer Algebra Handbook" by Grabmeier, Kaltofen and Weispfenning (eds.) (2003) includes some advanced topics in group theory and examples of code that you can use with GAP.
In ...
- 526
8
votes
How can I check whether a given finite group is a semidirect product of proper subgroups?
In general such a subgroup $H$ is called a complement to $N$. Complements could be conjugate, and so GAP has a function ComplementClassesRepresentatives that ...
- 17.6k
8
votes
How do Gap generate the elements in permutation groups?
I will just show how to find this information in GAP - a skill that may be useful in a number of situations.
GAP has is a function PageSource which can show the ...
- 6,952
8
votes
Accepted
Is this group representation faithful?
The answer is that the group is isomorphic to an extension of ${\mathbb Z}^3$ by $A_4$, but it is a nonsplit extension, and not a semidirect product. Your representation must be faithful, because no ...
- 87.8k
8
votes
Accepted
Construct a semidirect product in GAP
What you are providing is a perfectly good description of the product as far as a textbook is concerned. Alas, GAP is really picky in that objects are not just "somehow" as you describe, but very ...
- 17.6k
8
votes
Accepted
In GAP, How can I check whether a given group is a direct product?
There is the command StructureDescription(G) which gives a name for the group $G$ and could tell you whether $G$ is a direct product or not. However, it just gives ...
- 25k
8
votes
Accepted
Finding low-index normal subgroups of finitely presented groups in GAP
I implemented a GAP package that provides an algorithm
for computing normal subgroups of a finitely presented group up to a given index bound. (This algorithm is based on the thesis of David Firth at ...
8
votes
Accepted
An explicit representation of a free nilpotent group by unitriangular matrices
I think that I have found an embedding of $F(3,2)$ into ${\rm UT}(5,{\mathbb Z})$. I used a combination of GAP and Magma to do this, but here is the (putative) result in GAP.
The images of the three ...
- 87.8k
8
votes
Accepted
Structure of the Brainball group
I'm not sure whether your computations are enough or not, so I'll refrain from answering that.
Instead, here's an alternative approach to show your group $G$ is isomorphic to the wreath product $W := ...
- 6,565
8
votes
How do i see how all the generators get mapped during automorphism in GAP
The notation ^perm denotes an inner automorphism -- conjugation by the indicated permutation. The result tells you that $S_3$ has no outer automorphisms.
If you ...
- 17.6k
7
votes
Accepted
Rubik's Revenge Cube in GAP
It depends on the definition of the puzzle: If the faces are colored uniformly, for example a swap of 5 and 6 is not seen in the puzzle. You can see this discrepancy in the group. We construct the ...
- 17.6k
7
votes
How many groups of order $2058$ are there?
I believe the answer is 91.
This is calculated by looking at where the construction process hangs (conjugacy test of small cyclic subgroups with many regular orbits) and reducing this test for ...
- 17.6k
7
votes
Accepted
Minimal Permutation Representation Degree of a group: GAP implementation
Added 1/23:
There now (GAP 4.12) are built-in functions MinimalFaithfulPermutationDegree and ...
- 17.6k
Only top scored, non community-wiki answers of a minimum length are eligible