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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What group is this? [closed]

What type of group could this be? I need help with identifying it. It is a subgroup of $S_{50}$ With the following 10 generators: a=(7, 43)(11, 42)(21, 30)(26, 32)(29, 33), b=(5, 50)(8, 20)(17, 48)(...
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1answer
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Find all homomorphisms from a Finite Field (using GAP)

I'm currently working with $\mathbb F_{243}$, viewing it as an additive group. I'd like to have GAP calculate all of the (additive) homomorphisms from this group to smaller groups. I've done this with ...
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Real roots of a polynomial in GAP, and RootsFloat function

Is there a way in GAP to calculate the roots of a polynomial, say $x^2-2$, as real numbers? I know I can express the roots as cyclotomic numbers, but then (real) cyclotomic numbers are not ordered as ...
3
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1answer
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Wedderburn decomposition of commutative finite dimensional algebras via GAP

Let $R$ be a commutative finite dimensional $K$-algebra over a field $K$ (for example the monoid ring of a a finite monoid over a field). Assume we have $R$ in GAP. Then we can check whether $R$ is ...
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2answers
67 views

Finding all small monoids with the help of GAP

In GAP there is the command AllSmallGroups(n) to construct all finite groups of order $n$ up to isomorphism. Question: Is there a similar method (or package) in ...
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1answer
40 views

Defining the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in GAP

I am looking to define the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in the software GAP. In general, one can define these groups separately. For example, let's say ...
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How to best input and work with matrices over GF2 in GAP?

I am working on a problem where I have a group of matrices and a vector space of vectors, and I am looking to calculate the orbits of said group upon said vector space with GAP. Right now, I have each ...
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4answers
220 views

In GAP, how can I create a group isomorphic but not equal to a given group?

Sorry if the question title is confusing. Let me try to explain my question over an example. Please consider gap> G := SmallGroup(24, 12);; Without knowing that ...
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Weyl group extension as inner automorphisms

Notations: $L$ a semisimple Lie algebra over $\mathbb{C}$ with $\{e_i,f_i,h_i\}_{i=1}^{n}$ the algebraic generators $H$ the Cartan subalgebra $W=\langle s_i|i=1,\cdots,n\rangle$ the Weyl group and $A=...
2
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1answer
62 views

Conjugacy classes in GAP

Let $V(FA_4)$ be normalized unit group of group algebra $FA_4$, where $F$ is a field containing 4 elements and $A_4$ is alternating group on $4$ symbols. How can I find conjugacy classes of elements ...
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1answer
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How to find out if a semilinear representation is irreducible (possibly with gap)

Let be $\Gamma=\Gamma(2^6)$ the semilinear group on $GF(2^6)$, namely the group of semilinear mappings $\tau_{a,\sigma}\colon x\to ax^\sigma$ for $x \in GF(2^6)$ as a vector space over $GF(2)$, $a \in ...
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1answer
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How to construct submodules with GAP / MeatAxe?

Let $G= \langle g_1, g_2 \rangle$ be a finite group. Let $k$ be a finite field with ${\rm char}(k)=p>0$ such that $p \mid |G|$. Let the $kG$-module $M$ be a MeatAxe-module in GAP. The generators of ...
3
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1answer
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Computing Group Extension and $2$-cocycle in GAP

It is a well known theorem that $2$-cocycle are in bijection with Group extensions. Suppose if I have group extension, say, for example, $$1\rightarrow C_2 \rightarrow D_{16}\rightarrow D_8\rightarrow ...
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1answer
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Is there an OpenSource CAS compatible with GAP where relatively fast computation of Hom$_{kG}(M,N)$ over "large" finite fields is possible?

I'm working in the area of modular representation theory of finite groups ($G$ is a finite group, $p$ is a prime number dividing $|G|$, $k$ is a finite field of characteristic $p>0$, $M$ and $N$ ...
2
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1answer
64 views

Defining a group in GAP without having its presentation

I know how to define a group when we have its presentation in GAP by using FreeGroup command over the generators and then taking quotient over the relators, however what if I only have the group as a ...
5
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2answers
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A conjecture about non-nilpotent groups

A finite group, $G$, is nilpotent if its upper central series terminates (at $i \in \mathbb{N})$ with $Z^i(G)=G$, where $Z^i(G)$ is it's $i$-th center which can be described as $\{x \in G\mid \forall ...
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1answer
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More convenient GAP code to verify Additional property of d-maximal groups

Let $G$ be a finite $p$-group and $d(G)$ be its minimal number of generators. We say that $G$ is $d$-maximal if $d(H) < d(G)$ for all $H < G$. The following code determines weather $G$ is $d$-...
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1answer
75 views

Computing Hochschild cohomology groups $H^i (A,M)$ in GAP/QPA

Let $A$ be a finite dimensional $k$-algebra. I would like to calculate second Hochschild cohomology group $H^2(TA,DA)$, where $TA$ is the trivial extension of $A$, and $DA = Hom_k (A,k)$. For example <...
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0answers
90 views

How to solve A*x=B in GAP (A,B are matrices)

Solving $A x = b$ with matrix $A$ and vector $b$ can be done in GAP using SolutionMat. How would you find the solution to $A x =B$ with $B$ a matrix? Are there simplifications if I only need to know ...
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0answers
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How to take the imprimitive unitary reflection groups as input in GAP?

I am wondering about how to take the group $G(m,p,n)$ as input in GAP. The groups $G(m,p,n)$ appear in the classification of unitary reflection groups. The group $G(1,1,n)$ is the symmetric group $S_n$...
3
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1answer
195 views

Programming (Computer Algebra System) for Paired Suborbits

The orbits of $G \leq Sym(\Omega)$ on $\Omega × \Omega$ are called the orbitals of $G$ on $\Omega$. For each orbital $\Delta$ there is a paired orbital denoted $\Delta'$, where $(y, x) \in \Delta'$ if ...
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1answer
79 views

How to generate random combinations from large set in GAP

I need to generate random combinations (unordered tuples without duplication). This code works fine for relatively small $n$ but I need much larger n $n>200$ for example ...
4
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1answer
80 views

Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices

I wish to determine the type of Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. I've been using GAP ...
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2answers
88 views

Faithful permutation representation

excuse me if my question is trivial. I’m trying to use magma to construct faithful permutation representations of a certain group using the group action that lets the group G acts by the left ...
2
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1answer
50 views

How GAP system implements Todd-Coxeter algorithm

I know that GAP uses Todd-Coxeter algorithm to form the coset table of a group but there are multiple options while forming the table using this algorithm, so I was wondering if there is any way to ...
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1answer
53 views

Computing the Stabilizer of a coset $Nh$ in $H=N.G$ using GAP. [closed]

Let $H=N.G$ be an extension of $N$ by $G$ and $Nh$, $h\in H$, a coset of $N$ in $H$. How do one compute the stabilizer $S$ of $Nh$ in $H$ using GAP?
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1answer
65 views

How do i see how all the generators get mapped during automorphism in GAP

I'm using GAP for the first time. I used AllAutomorphism command to see all automorphisms of SymmetricGroup of order $3$. ...
5
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1answer
57 views

Constructing the Galois group of a polynomial knowing its splitting field as a matrix algebra

I started with the polynomial: $p(x) = x^6-6x^4-6x^3+12x^2-36x+1$ and obtained four 12x12 matrices that have the polynomial $p$ as a minimal polynomial. Using GAP I found out that the Galois group of ...
3
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1answer
64 views

How to obtain a certain set of matrices in GAP?

I'd like to ask the following question: Let $A$ be a finite set of complex numbers. I would like to obtain in GAP the set $B$ of all $n\times n$ - matrices where in each row and column there is ...
5
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1answer
174 views

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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1answer
150 views

GAP on Jupyter Notebook

I wanted to learn GAP as it is certain that it will be helpful in the future for my research. I wanted to try GAP in Jupyter notebook in GitHub. I have tried following the step i.e. launch binder and ...
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1answer
58 views

Projective General Linear Group in GAP

I know that in the GAP system the command $PGL(d,q)$ gives the projective general linear group of $d \times d$ matrices over the finite field with $q$ elements, But what if I need $PGL(d,F)$ where ...
4
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1answer
87 views

Semidirect product in GAP (Wallpaper groups)

I am fairly new to GAP and I am having difficulties using the semidirect product method. I am trying to use this as I am working with wallpaper groups. In my example, I am trying to figure out : $D_6 \...
2
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0answers
72 views

Help with GAP code on finding generators of spherical systems.

I am looking at a specific group $G$ and trying to find an index 2 subgroup with generators of length 3, such that the generators form a spherical system. Here, an $r$-tuple $[g_1,\ldots,g_r]$ of ...
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1answer
54 views

Simplification of a group presentation

Im new to MAGMA and hope somebody will help me with my question. If a group has a presentation with 4 generators, is there a magma code/function that can give me the same group with only three ...
2
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1answer
40 views

Identify conjugacy class of coset in factor group in GAP

I wish to use GAP to identify the conjugacy class of a given element $[g]$ in the factor group $G/H$ where $H$ is a finite-index normal subgroup of a finitely presented group $G$. I know how to ...
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0answers
46 views

Code for LCM of order of element in GAP

Is there an easier way to write the LCM part of this code? Gap only allows two arguments for LCM_INT. ...
1
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1answer
63 views

Finding a group with minimal generators and then a subgroup generated by these as an index two normal subgroup.

Given a group with seven generators and seven relations, each of length 3, how can I use GAP to find the group generated by only three of its generators? For example, $$G = \langle a,b,c,d,e,f,g \mid ...
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2answers
47 views

Changing the notation of a permutation in GAP

I have written a program in GAP which has output three permutation elements. An example is (1,3,4,2),(3,5),(2,4). What I would like is that the notation for the permutations includes all numbers, even ...
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1answer
52 views

Finding maximal quotient of a particular class of a group

Given a group G, how do I find the maximal 2 quotient of class 3 for that group in GAP? For example, if ...
2
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0answers
72 views

Finding special presentations for finite groups

Let $G$ be a finite group. Call a presentation of $G$ "normalised" (I do not know whether such presentations by generators and relations have been studied before and I invented the name &...
7
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1answer
99 views

Outer automorphisms of extraspecial groups in GAP

Let $G_n$ be the extraspecial group of order $2^{1+2n}$. Its outer automorphism group is known to be isomorphic to the general orthogonal group $GO(2n)$. I'd like to get an explicit map of this ...
3
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1answer
153 views

simultaneous diagonalization of set matrices

I have a set of integral square invertible symmetric matrices $A_i$ with $A_i^2=I$ (so also $A_i A_i^T=I$). The matrices commute. I'd like to map them simultaneously to a set of diagonal matrices $D_i$...
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1answer
77 views

Embedding a finite group into $\operatorname{GL}_n( \mathbb{Z})$ [duplicate]

Let $G$ be a finite group. Having a concrete realisation of this finite group as a group of matrices is often helpful for calculations, which motivates the following questions. Question 1: Is there a ...
2
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1answer
90 views

In GAP, how can I generate a specific diagonal subgroup of the direct product two groups?

Let $P$ and $Q$ be isomorphic subgroups of a finite group $G$, let $\phi$ be an isomorphism from $P$ to $Q$. How can I find the subgroup of DirectProduct(G, G) that corresponds this set $\{(a,b)\in G\...
2
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1answer
49 views

Finding a K-algebra of centralisers via GAP

Let $X$ be a finite set of $n \times n$-matrices over a field $K$. Let $A_X$ be the $K$-algebra of $n \times n$-matrices $Y$ with $YS=SY$ for all $S \in X$. Question: Is there a quick way to obtain ...
2
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1answer
81 views

orthogonalization in GAP

I'm using Eigenvectors in GAP to diagonalize a matrix $M$. The matrix is guaranteed to be diagonalizable and in fact it should be so by an orthogonal transformation. This works fine and $D$ is the ...
4
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1answer
100 views

Filtering Non-abelian groups of fixed order in GAP [closed]

I was learning GAP commands for operations on groups. I needed to consider non-abelian groups for simplicity in computation. The command Filtered is useful to find out abelian groups in the following ...
2
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0answers
200 views

Bruhat decomposition of parabolic subgroup from that of borel subgroup

For $G=Sp(n,q)$, I can take an arbitrary element $S$ and find its Bruhat decomposition according to the Borel subgroup (in this case lower triangular symplectic matrices). This is thanks to a ...

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