Questions tagged [gap]
GAP (Groups, Algorithms and Programming) is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. It provides a programming language, a library of thousands of functions implementing algebraic algorithms, and large data libraries of algebraic objects. Please note that GAP Forum or GAP Support may be more suitable places for questions about GAP: see http://www.gap-system.org/Contacts/Forum/forum.html
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Which resources are available to self-study GAP?
Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer ...
- 2,718
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How do Gap generate the elements in permutation groups?
I understand that permutationgroups in Gap are represented by generators, which seems to be far more efficient than groups represented by all it's elements, but how could then for example
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finding code of a function in GAP packages [closed]
How can I find the codes related to a function in GAP? When I use "??" in front of the name of function there is no help, so I want to find the code of function among packages.
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Proof that the Rubik’s Cube group is 2-generated
Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube:
Frank Barnes observes that the group of the cube is generated by two moves:
\begin{align*}
\alpha &= L^2 B R D^{-1} L^{...
- 3,248
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1
answer
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How to define own group action in GAP?
I am beginner in GAP. I have a group and a set. I wish to define an action the group on the set in my own way and wish to calculate its orbits and stabilizers. Is it possible? What is process?
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Smart way to solve octics like $x^8+5992704x-304129728=0$
Question: How to quickly find explicit formula for roots of polynomials (solvable by radicals) like $f(x)=x^8+5992704x-304129728$?
My current approach is just a brute force - not very fast and ...
- 552
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Easy way to get real irreducible characters (reps) from complex irreducible characters?
For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
- 2,155
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GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$?
In GAP, working with elements of the symmetric group $S_k$ is straightforward. E.g. one can write
(1,2)*(2,3);
to obtain (1,3,2).
Is there a similar ...
- 282
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votes
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answer
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Conjugacy classes and centralizers of a SmallGroup
What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)?
Would someone be willing to provide the complete lists of conjugacy classes and centralizers of SmallGroup(...
- 3,607
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Semidirect Products with GAP
I'm wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I'm trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. ...
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Is $\mathrm{gnu}(2304)$ known?
(Here, $\mathrm{gnu}(n)$ denotes the number of groups of order $n$ up to isomorphism.)
I wonder whether the number of groups of order $2304=2^8\times 3^2$ is known. GAP exited because of the memory. $\...
- 83.1k
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How to find presentation of a group using GAP?
I have a group from Small Group Library and I want to find its presentation using GAP.
I have tried to use PresentationFpGroup(G) but failed.
Please suggest me a method.
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How to express all group elements in terms of generators in GAP
I have a permutation group given in terms of its generators, such as
G := Group((2, 3), (1, 4), (1, 3)(2, 4));
I can iterate through the group and get all ...
- 304
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How is GAP generating all subgroups?
GAP can return all subgroups of $S_4$ pretty much instantaneously:
...
- 585
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How to check in GAP whether two groups are isomorphic
Let $G,H$ be two Groups. How to check with GAP whether they are isomorphic or not?
For example, GAP has IsNilpotentGroup to check whether the group $G$ is ...
- 799
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A property for some finite groups (especially ${\rm PSL}(2,13)$)
Let $G$ be a finite group of order $n$ and consider the following property:
(P) for every factorization
$n=ab$ there exist subsets $A$ and $B$ such that $|A|=a$, $|B|=b$ and $G=AB$.
($AB=\{ab:a\in ...
- 2,507
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votes
1
answer
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GAP semidirect product algorithm
Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group?
EXAMPLE:
...
- 738
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Factoring a polynomial in $\mathbb Q[t][x]$ over an extension of $\mathbb Q[t]$
I have found the following unit-distance embedding of the cubic symmetric graph $F_{42}A$ with $D_7$ symmetry. (Gerbracht's embedding on the MathWorld page for cubic symmetric graphs is only of $C_7$ ...
- 102k
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Differences between GAP and PARI/GP?
I was looking for free alternatives to Magma to perform number theory computations, such as modular, factorization, etc.
and I've found GAP and Pari.
What's the difference between them?
Any other ...
- 389
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1
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Use GAP to define finite Clifford group
I am studying the computer algebra system GAP to do some calculations about Clifford group, which is defined (cf. Lawson and Michelsohn, Spin Geometry, Princeton 1989) as followings
Definition. Let's ...
- 63
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Computing Conjugacy Classes of Subgroups in GAP
GAP has the command ConjugacyClassesSubgroups which gives a list of the conjugacy classes of a finite group $G$. Is there a way I can specify further what types of subgroups GAP reports? For ...
- 31.3k
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A normal intermediate subgroup in $B_3$ lattice?
Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$.
An intermediate subgroup $H \subset K \subset G$ is a ...
- 5,787
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2
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How can I calculate $gnu(17^3\times 2)=gnu(9826)$ with GAP?
I tried to calculate the number of groups of order $17^3\times 2=9826$ with
GAP.
Neither the NrSmallGroups-Command nor the ConstructAllGroups-Command work with GAP. The latter one because of the ...
- 83.1k
4
votes
1
answer
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Commutator of two elements in group algebra $\mathbb F_{5}D_{30}.$
I want to understand how to find the commutator of two elements in the group algebra $\mathbb{F}_{5}D_{30}$ using GAP. Additionally, I would like to determine the nilpotency class of the nilpotent ...
- 5,932
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Can graphs in GAP be obtained as a visual output
In GAP when we draw a cayley graph of a group using "CayleyGraph" command we get a list. Is there a way to visualize that cayley graph like a figure?
Or else is there any other software that has the ...
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2
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Subgroup lattice of UT(3,3)
I need to draw a subgroup lattice of $ UT(3,\mathbb{Z_{3}}) $, the group of upper triangular matrices with diagonal one.
How to do it? And whether there is somebody ready image?
(I know that it can ...
2
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1
answer
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Matrices' outputs in GAP
Does anyone know whether is a function in GAP via I would be able to change elements of matrices over GF(2) to the form 0 and 1. need to say, in GAP, outputs for elements of matrices defined over GF(2)...
- 1,298
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Websites/Software for Group computation
Anyone knows a website or software that helps to do computations in a group? For example, by inputting generators and relations in the group, can we tell when two particular elements in the group ...
- 5,263
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How does one define a representation in GAP?
It seems that the reference manual for GAP never mentions how to construct a representation by sending generators to specified matrices. Yet I cannot help but feel that such a basic task must be ...
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How to make GAP understand that a homomorphism is a representation?
In a previous question I asked how one constructs an explicit representation in GAP. As a concrete example, I want to consider the $5$-dimensional $\mathbb{F}_3$-representation $\rho$ of $C_8$ sending ...
- 89
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GAP Isomorphism between two finitely presented p-groups
I have finitely presented nilpotent p-groups with exponenta 5 G = $\langle x1, x2, x3, x4: [x1, x2] = 1, [x3, x4] = 1, [x1, x4] = [x3, x2], [x2, x4] = [x1, x3]^{-2}] \rangle$ and H = $\langle x1, x2, ...
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How to solve this solvable 8th-degree algebraic equation by radicals?
Solve the following equation in radicals.
$$x^8-8x^7+8x^6+40x^5-14x^4-232x^3+488x^2-568x+1=0$$
I use Magma to verify that its Galois group is a solvable group.
...
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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)?
A derangement is a permutation that has no fixed points.
My question is . . .
What is known about subgroups of a symmetric group $S_{n}$ that contain only derangements (plus the identity)?
It is ...
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How many groups of order $2058$ are there?
I tried to calculate the number of groups of order $2058=2\times3\times 7^3$ and aborted after more than an hour. I used the (apparently slow) function $ConstructAllGroups$ because $NrSmallGroups$ did ...
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List Conjugacy Classes in GAP?
I have a permutation group $G$, a subgroup of $S_{81}$, defined by a set of 6 specific permutations.
This permutation group has order 2592, and 54 Conjugacy Classes.
I can obtain a class ...
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How many groups of order $512$ and $1024$ are there with a center of size $2$?
I did not find a sequence in OEIS about the number of groups of a given order with a center of size $2$.
For the first few powers of $2$, the numbers are :
$2$ : $1$ group
$4$ : $0$ groups
$8$ : $2$ ...
- 83.1k
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What does "pc group" mean in GAP?
For example, when I enter G:=CyclicGroup(2); in GAP, the program returns with the information ...
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What free software can I use to solve a system of linear equations containing an unknown?
Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$?
Presumably I could do this ...
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Find generators of a group in GAP
This is a question in the mathematical software called GAP:
What is the command for displaying all the generators of a given group?
I have been searching around but yet not found anything helpful, ...
- 4,445
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Perfect groups whose character degrees square divide its order
This post is an extension of that one from the non-abelian finite simple groups to the finite perfect groups. According to the mentioned post and its second comment, for all non-abelian finite simple ...
- 5,787
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votes
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answer
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GAP Most efficient way to check multiple properties of a group in the small group library
In GAP I would like to search the small groups library looking for groups with specific properties (I suppose this is the most common usage).
If I have a list of properties I want to test, what is ...
- 1,532
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Presentation for hand calculation
I am looking for a presentation of SmallGroup(32,7) which is more amenable to hand calculation. More precisely, I used the following comments to find a ...
- 527
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answer
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Reconstructing a polynomial's Galois group from those of its subfield factors
Having worked out that the Galois group of
$$p(a)=a^{4} \left(16 t^{4} + 16 t^{2}\right) - 64 a^{3} t^{3} + a^{2} \left(- 8 t^{4} + 24 t^{2}\right) + 16 a t + t^{4} + 5 t^{2} - 4$$
over $\mathbb Q[t=\...
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Sums of products of average character values on cosets
Consider a finite group $G$, a subgroup $H\leq G$, and a transversal $G/H = \{t_1H, t_2H,\ldots,t_rH\}$. Given three characters $\chi_1,\chi_2$ and $\chi_3$ of $G$, I'd like to compute:
$$
\sum_{i}^r ...
5
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2
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Graphical interface in GAP
Is there any graphical interface in GAP? Something like RStudio for R or WxMaxima for Maxima. I'm using GAP under a Linux system.
Thanks
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Minimal Permutation Representation Degree of a group: GAP implementation
For a finite group $G$, let $\mu(G)$ be the least positive integer $n$ such that $G$ is embedded as a subgroup of the symmetric group on $n$ points. In other words, $\mu(G)$ is the minimal permutation ...
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Algorithm for Finitely Presented Torsion-Free Nilpotent Groups
I am studying some finitely presented, torsion-free and nilpotent groups $G$ and need to consider the following question:
Let $H$ be a subgroup of $G$ and suppose that $H$ is generated by $h_1,\cdots,...
- 5,263
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votes
1
answer
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How do I substitute a value into a polynomial in GAP?
Question: How do I substitute a value into a polynomial in GAP?
So, if I start off with the following:
x:=Indeterminate(Integers,"x");
f:=x^2+3;
I have $f$ as ...
5
votes
2
answers
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Are the primitive groups linearly primitive?
A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup.
A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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