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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 ...
Oscar Lanzi's user avatar
  • 41.1k
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....
ahulpke's user avatar
  • 18.8k
13 votes
Accepted

Sylow $2$-subgroup Mathieu Group $M_{24}$

While I would emphasize Derek Holt's remark on that "identification" does not make much sense, here is a way that shows the Sylow subgroup to be isomorphic to the semidirect product of an ...
ahulpke's user avatar
  • 18.8k
12 votes
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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 ...
Olexandr Konovalov's user avatar
11 votes
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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/...
verret's user avatar
  • 6,769
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 ...
Lee Mosher's user avatar
  • 124k
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 ...
sTertooy's user avatar
  • 6,205
9 votes
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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?...
ahulpke's user avatar
  • 18.8k
9 votes
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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, ...
ahulpke's user avatar
  • 18.8k
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}{...
ahulpke's user avatar
  • 18.8k
9 votes
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Minimal Permutation Representation Degree of a group: GAP implementation

Added 1/23: There now (GAP 4.12) are built-in functions MinimalFaithfulPermutationDegree and ...
ahulpke's user avatar
  • 18.8k
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 ...
Olexandr Konovalov's user avatar
9 votes

Sylow $2$-subgroup Mathieu Group $M_{24}$

Maybe there is a beautiful presentation of $P\in Syl_2(G)$, but Magma chucked a few out and I chose the smallest. It is generated by four elements, $a,b,c,d$, subject to the 12 relations $$a^4 = b^4 = ...
David A. Craven's user avatar
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 ...
Derek Holt's user avatar
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8 votes
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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 ...
ahulpke's user avatar
  • 18.8k
8 votes
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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 ...
Mikko Korhonen's user avatar
8 votes
Accepted

Use GAP to define finite Clifford group

If you look at the definition, you'll note that they use "-1" as an extra generator that is central and of order 2. In GAP you can't call it that way, thus lets call it ...
ahulpke's user avatar
  • 18.8k
8 votes
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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 ...
Friedrich Rober's user avatar
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 ...
Derek Holt's user avatar
  • 92.1k
8 votes
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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 := ...
sTertooy's user avatar
  • 6,205
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 ...
ahulpke's user avatar
  • 18.8k
7 votes

In GAP, How can I check whether a given group is a direct product?

The question would be more interesting if you asked how you might do this rather than asking whether there is a command to do it in GAP. Here is a first suggestion, which you could easily implements ...
Derek Holt's user avatar
  • 92.1k
7 votes
Accepted

Monster coefficients

In GAP, you could simply iterate in a triple loop over $\phi,\psi,\rho$, calculate the $g$-values and find maximum and sum values: ...
ahulpke's user avatar
  • 18.8k
7 votes

Can graphs in GAP be obtained as a visual output

A standard tool for visualizing graphs is the graphviz software. The following GAP program ...
ahulpke's user avatar
  • 18.8k
7 votes

Subgroups of $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$

GAP is open source. You can download and install it as explained here. Then, for example, you can do the following. Create the group in question: ...
Olexandr Konovalov's user avatar
7 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. For example, if $G$ is any non-abelian simple group, then $G/\langle\langle g\rangle\rangle$ is trivial for every $g \neq e$, but $G$ is not abelian.
N. S.'s user avatar
  • 133k
7 votes
Accepted

how to save a file in GAP

LogTo(filename) will just print everything that you see in your terminal to a file (including output and errors), so you won't be able to read it in. If you want to store your input to read in later, ...
ChockaBlock's user avatar
7 votes
Accepted

Computing endomorphism ring of finite groups via computer

In GAP, AllHomomorphismClasses is (and I'm writing this as the person who wrote the function) a "toy" function that is provided to try small examples in ...
ahulpke's user avatar
  • 18.8k
7 votes
Accepted

Filtering Non-abelian groups of fixed order in GAP

Expanding my comment above to the proper answer: The input S:=Filtered(AllSmallGroups(16), G -> IsAbelian(G)=false); is syntactically correct, and it actually ...
Olexandr Konovalov's user avatar

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