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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GAP — null multidimensional array

I need to create an $n\times a \times b$ array in GAP. How do I do that fast? I tried putting ...
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GAP — GeneratorsOfRing giving a list with repeated element

I put the following code in GAP: R := Integers mod 8; and I get the answer: [ ZmodnZObj( 1, 8 ), ZmodnZObj( 1, 8 ) ] Can ...
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GAP — make an Additive Group into a Group

I have a list of vectors $\texttt{A}$ with coordinates in $\{0,1,\ldots, n-1\}$ that happen to make a group under addition. How do I make GAP understand that it is a group? One possible solution that ...
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Basis for $\operatorname{Ext}^1(M,N)$ in QPA

It is possible to calculate for two given modules $M$ and $N$ $\operatorname{Ext}_A^1(M,N)$ using the GAP-package QPA. Assume $\operatorname{Ext}_A^1(M,N)$ is n-dimensional. Question: Is there a ...
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How to compute the stabilizer subgroup of a partition with GAP?

A partition $P$ of a set $S$ is a set of disjoint subsets of $S$ whose union is $S$. Let $G$ be a subgroup of the symmetric group $S_n$. Define the stabilizer subgroup of $G$ for a partition $P$ of $\{...
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GAP – compute kernel of a matrix with coefficients in a finite ring

I need to compute first cohomology group with coefficients in $(\mathbb{Z}/n\mathbb{Z})^m$ of specific finite groups. I reduced the computation of cocycles to the following problem: compute the kernel ...
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Computing a formula in either MAGMA or SAGE or GAP? [closed]

Fix the positive integer numbers $t_1, t_2, t_3,t_4, t_5.$ We have the following formula: $S = \sum\limits_{i,\, j,\, h,\, m,\, k_1 + k_2+k_3+k_4 = i-t_1,\, \ell_1+\ell_2 + \ell_3 = j -t_2 + k_4,\, ...
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Computing the multiplication of elements to generate the Cayley graph of a semidirect product

I have computed a semidirect product, $s$ of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ as below and have drawn a Cayley graph for $s$ with respect to a generating set $S$. But I wanted ...
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In how may ways can we remove lines such that all dots will still be connected?

Let there be a square, connected with 9 dots and 12 lines. We can select some lines (at least 1, maximum 4) among the 12 lines, and then remove them. In how may ways can we remove lines such that all ...
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Algorithm for finding all sets that fulfill a condition?

Lets say I have a set of none-negative integers of given length $n$ where $n>0$. $$m=\left(m_1,...,m_n\right)$$ A number can appear more than once in the same set. The same set or numbers can also ...
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How to obtain the product of elements in a semidirect product using GAP

I have computed a semidirect product of $\mathbb{Z}_5 \times \mathbb{Z}_5$ by $\mathbb{Z}_3$ in GAP as shown below. If I need to obtain the product of two elements of the semidirect product group $(\...
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Quiver and relations for a quotient algebra via QPA

Given a finite dimensional quiver algebra $B$ and a basic idempotent $e$. Is there a quick way to obtain the algebra B/BeB by quiver and relations using the GAP-package QPA? One way to do this is to ...
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GAP: how to obtain the Young Symmetrizer?

Given a partition $\lambda$ of $n$ and a standard Young Tableaux filled with numbers from $1$ to $n$ (e.g. increasing row by row), how does one obtain the corresponding Young Symmetrizer using GAP? ...
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Understanding Direct product of subgroups in GAP

I am trying to obtain a direct product in GAP, but I am not able to understand the output. The ploblem is the following, I have a GL(2,3) group of 48 elements ($2\times 2$) matrices, and I want to ...
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Images of the generators of $D_{10}$ under its automorphisms.

I have constructed the dihedral group generated by $a$ and $b$ of order $10$ in GAP by the following way: ...
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Dihedral group of order 10 in GAP

The dihedral group of order $10$ is given by $D_{10} = \langle a,b| a^5 = b^2 = 1, bab^{-1} = a^{-1}\rangle$. Now I need to find all the elements in GAP. But whenever I type ...
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How to manually write an element of group ring in GAP. [duplicate]

Suppose, we consider R:=GL(29); (Galois field of order $29$) G:=SmallGroup(7,1); (cyclic group of order $7$) H:=Group Ring(R, G); (this generates the group ring). Suppose that GAP is ...
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49 views

What are those special orbits of a group acting on a direct product of several copies of a normal subgroup?

Suppose $G$ is a finite group and $N$ a normal subgroup whose order $n$ is coprime to its index $m$. I start with an arbitrary section $\sigma: G/N \rightarrow G$. Then each $g \in G$ can be written ...
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is it possible to chose an element of a group without invoking the complete list of elements of a group using GAP?

Suppose $G$ is a group of finite order. Then the command L:=List(G) in GAP gives us the complete list of elements of G and we can choose any of the element using the command L[i]. Now is it possible ...
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How do I construct the group algebra of a group in GAP?

I tried the following: ...
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Deleting relations in GAP/QPA

Given a quiver algebra $A$ (example A:=NakayamaAlgebra([4,4,3,2,2,1],GF(3)) )in QPA, one can get the relations of A as follows: rel:=RelatorsOfFpAlgebra(A); In the example the output is [ (Z(3)^0)*...
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Going back and forth between algebra $A$ and regular module ${}_AA$ in GAP

Let A be an algebra over some field (e.g., Rationals) and L := LeftIdeal(A, [g]); be a left ...
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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 ...
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Explicit description of the conjugation action of $[[1,0],[0,p]]$ on the amalgam $SL(2,\mathbb{Z})*_{\Gamma_0(p)} a_pSL(2,\mathbb{Z})a_p^{-1}$

Let $a_p$ denote the matrix $[[1,0],[0,p]]$, where $p$ is prime. Then $SL_2(\mathbb{Z}[1/p])$ can be presented as the amalgamated product $$SL_2(\mathbb{Z})*_{\Gamma_0(p)} a_pSL_2(\mathbb{Z})a_p^{-1}...
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Finding the image of an element under an automorphism by GAP

I am doing GAP to find the image of an element under an automorphism of a certain group: ...
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Finding right transversals using GAP

I am new to the GAP software and I am following the GAP manual. I have found one right transversal to a subgroup of a group. Now i want to know if we can find all the right transversals of a subgroup ...
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1answer
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How to know if an automorphism is induced by the normalizer

This is directly related to an initial question I had here. I want to followup this question with another one. Supposing I know $G\le S_n$ is a permutation group and that $C_1$ and $C_2$ are conjugacy ...
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In $\mathbb Z_{p^k}$ solve $(ap+b)^{p-1} = 1 $ for $a$, given $0 < b < p$, and $p$ an odd prime.

In the ring $R =\mathbb Z_{p^k}$ we can give to each element $g \in R$ two "coordinates" $a$ and $b$ as follows: $b = g \;\operatorname{mod} \; p$ and $a = (g-b)/p$. So we have in a unique fashion ...
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Proof of $(px+1)^{p^{k}}=(p^{k+1}x+1)$ mod $p^{k+2}$ for any odd prime $p$

I need to prove this identity using only simple arithmetics and combinatorics (or to find a counter-example): $(px+1)^{p^{k}}=(p^{k+1}x+1)$ mod $p^{k+2}$, which I verified using several values of $p$ ...
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1answer
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Generate a group with Cyclotomic representation in GAP

I am trying to create a matrix group in GAP, but I am having problems because "it runs out of memory". The group in question is related to GL(2,3). The problem is that I do not know if I am ...
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GAP code to calculate the a certain subgroup $E(G)$ of a group

I am a research scholar from India. At present, I am working on a problem. For this problem, I need to construct the subgroup $E(G)$ of a group $G$ in GAP. Please help me. My question is as follows: ...
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1answer
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Construction of a group in Magma

I need to construct the following group in Magma: given $H=(C_2)^3 \rtimes (C_7 \rtimes C_3) \times C_3$ (so $\operatorname{SmallGroup}(168,43)\times C_3$), there is a non-split central extension by $...
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1answer
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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 ...
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A group theoretic interpretation of Lagarias inequality

Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
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1answer
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how to check if there is an automorphism mapping between two conjugacy class

Let $G\le S_n$ be a permutation group and suppose that $C_1,C_2$ are two distinct conjugacy classes that have the same cardinality and is represented by a permutation of the same cycle-type. My ...
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1answer
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Presentations of subgroups of S6 as permutations

Computer science professor, self-taught abstract algebraist. Beginner with GAP and SAGE. Can someone show me the quickest way to, when given the description of a subgroup of S6, obtain its ...
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2answers
95 views

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

Is there some program or a location which would allow me to work and calculate with the subgroups of the group $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$?
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1answer
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Identifying the group in GAP

I am defining a matrix group in GAP. I know that its a finite group, and can compute its order. Using sonata package and commands like ...
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How to solve decics like $x^{10}+100x^2+160x+64=0$ having Galois group 10T33?

Using the approach described in Smart way to solve octics like $x^8+5992704x-304129728=0$ (the method DecomPoly available in GAP) the decic quadrinomial from this ...
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Finite Fields in GAP. How to get $d$ from the finite field with $p^d$ elements?

I'm currently working in an algorithm to decompose matrices in classical standard generators. The thing is that the parameter of my main function gets a matrix m. ...
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Conjugacy Classes of Subgroups with Special Property for a Large Group in GAP

I am trying to do the following in GAP: $G$ is a given group of order $2^{19}$ and $H$ is a subgroup (but not normal) of $G$ of order $2^{15}$. My aim is to find a representative, say $A$, for each ...
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Generating words in a finitely presented group in SAGE

I'm trying to get a list of all words of length $n$ (in the word metric sense) in some finitely presented group. I have tried some very naive enumerations but it is very slow. Is there an efficient ...
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Schur Index for Quaternion Algebra

I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex". But I am working on the field $K=\mathbb Q (\sqrt{-39})$ ...
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1answer
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Need some help in generating the group element by generator using GAP system [closed]

I am new to GAP system. I would like to generate group using presentation of group. For example, dihedral group D_{2n} with generator . Can anyone familiar with GAP system help me out with this?
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Constructing Invertible 0-1-matrices [closed]

I am looking for effective algorithmic solutions for the following two problems: Question 1: Is there a quick way to obtain for a given $n$ the list of all integer $n \times n$ matrices having only ...
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107 views

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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1answer
60 views

GAP semidirect product

I am newbie in the GAP and in the group theory. Now I am trying to make semidirect product if GL(3,2) and GL(3,2) inversed and transposed. I use code below ...
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Galois field elements as Integers in $\mathbb{Z}$ [closed]

In the case of $GF(3)$ its elements are gap> Elements(GF(3)); of which I get [ 0*Z(3), Z(3)^0, Z(3)]. Further, as an example, to get the integer value of Z(3), I use gap> Int(Z(3)); of which I get 2....
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1answer
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Computing the class-preserving automorphism group of finite $p$-groups

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
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1answer
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Possible orders of trace k elements in $SL_2(\mathbb F_q)$

As a continuation of this question I would like to ask about possible orders of trace $k$ elements in $SL_2(q)$. Here are examples which I know. When trace is zero then we have $x^2=-1$ so it means ...