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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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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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1answer
54 views

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
36 views

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 ...
4
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1answer
80 views

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
47 views

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
55 views

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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0answers
78 views

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
37 views

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
71 views

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
89 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
69 views

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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2answers
93 views

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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0answers
29 views

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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33 views

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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39 views

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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0answers
26 views

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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1answer
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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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1answer
94 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
57 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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2answers
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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
120 views

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
37 views

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 ...
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1answer
69 views

Constructing minimal supplements with GAP

The problem is this: Suppose we are given a finite group $G$ and a normal subgroup $N$ of $G$ which is not contained in $\Phi(G)$. Call a subgroup $H$ of $G$ a minimal supplement for $N$ (in $G$) ...
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1answer
30 views

Store GAP REPL representation of expression in string

I have noticed that the representation that is displayed by the GAP REPL (read-evaluate-print loop) when an expression is evaluated sometimes differs from the result of printing that same expression, ...
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0answers
36 views

Transport/flow network in GAP

How to define a transport network in GAP? I can't seem to find a single function related to transport networks or flows in general in the GAP-GRAPE manual. I'd be content with just manually adding ...
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1answer
97 views

Presentation of $A_5 \times \Bbb Z_2$.

I want presentation of the group $A_5\times \Bbb Z_2$ which is a group of order $120.$ I know the presentation of $A_5$ but not of product. I tried it in GAP . In GAP its Atlas name is $2\times A_5$ ...
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0answers
59 views

Elements $A\in GL_4(\mathbb{Z}_2)$ with $A^5=I, A\neq I$ by using GAP.

I need elements $A\in GL_4(\mathbb{Z}_2)$(General linear group of $4\times 4$ matrices over $\mathbb{Z}_2$ ) with $A^5=I, A\neq I.$ By using simple calculation its hard to find such types of elements....
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1answer
66 views

To find Brauer Character for the special linear group SL(2,5) in GAP.

As suggest by Alexander Konovalov I load LoadPackage("ctbllib") and them worked for symmetric group $S5$. It worked for me as ...
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1answer
52 views

Product of $e \in A/J(A)$ with an element of $J(A)$ [GAP]

I have defined a matrix algebra over the rationals, calculated its radical, and took the semi-simple coefficient using h:=NaturalHomomorphismByIdeal and $ImagesSource(h)$. Afterwards, I calculated the ...
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2answers
74 views

Brauer Character in GAP.

What is command to obtain Brauer Character in GAP? In magma, it is like ...
6
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1answer
197 views

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 ...
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2answers
73 views

Linear independent subset of a set of vectors [GAP]

I'm trying to find a linearly independent subset of a set of vectors in a vector space over the rationals. Looking up "independent", "linear independence", or anything of the sort hasn't helped, since ...
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0answers
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GAP Integrated Development Environment

Is there any fancy editor of GAP code or even some environment dedicated to it? I've been using Notepad++ for this purpose but it seems only to color the syntax. I would like to have more ...
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0answers
20 views

WedderburnDecomposition of $FG/J(FG).$

WedderburnDecomposition(GroupRing(GF(p),G)) gives complete wedderburn decomposition for semisimple group algebra $FG$ for finitie group $G$ and finitie field $F$. But if $FG$ is not semisimple then ...
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1answer
65 views

jacobson radical nilpotence, associative algebras.

In $Pierce, Associative Algebras$, it is mentioned that the Jacobson radical of an associative algebra $A$ is nilpotent, meaning $J(A)^k=0$ for a natural number $k$. However, my professor said that ...
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0answers
104 views

What are some nontrivial things one could do with GAP? [closed]

I'm learning algebra at the undergraduate level and came across the group theory software called GAP. Despite some googling I'm not sure what things one could do with such software that would really ...
2
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1answer
34 views

Define the image of a representation of a finitely presented group in GAP

Please note that the example below is just a baby case in which my problem occurs. Of course there is a simple workaround for my problem in this particular example, but I would like to apply this to ...
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1answer
48 views

Computing quotients of group by elements of its lower exponent$-p$ central series

Let $G$ be a finite $p−$group of number of generators $d$ and exponent$−p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G)=1$ in the series $$ G=P_0(G)≥...≥P_{i−1}(G)≥P_i(G)≥... $$ ...
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1answer
30 views

Conjugacy class of a tuple in GAP

I would like to know how to obtain the conjugacy class of some tuple in GAP, or how two know if two tuples are conjugate by an element of some group (permutation group, more especifically). I know ...
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0answers
40 views

Conjugacy classes of stabilizer subgroups

Let $G\subseteq GL(V)$ be a complex finite reflection group. I would like to understand the stabilizer subgroups of $G$, and their normalizers. By this I mean (the conjugacy classes of) subgroups $H&...
2
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1answer
42 views

Network analysis of category of groups

I am taking a course in Network Analysis at my university and I would like to do something related to category theory in my home project. To be more precise, I would like to take some amount of finite ...
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85 views

Computing quotient cayley graphs in GAP

1) $X1$ is a cayley graph and a is an automorphism (an element of Aut($X1$)) of $X1$. I'm trying to compute the quotient graph $X1$/a in GAP. I tired to use the command "QuotientGraph(X1,[a]);" in GAP ...
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1answer
34 views

Is there a list containing regular graphs in GAP?

I'm trying to write a function in GAP (this is the first time I am using it) and I want to loop through like the "first" $1000$ non-complete, non-empty regular graphs. Do I have to construct this ...
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0answers
70 views

Identifying result from a Cayley graph in gap

I constructed a group s which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ using GAP commands as below. ...
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1answer
61 views

Interpreting semidirect product in GAP in terms of underlying groups

I obtained the group $G$ which is the semidirect product $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ in GAP as below. ...
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2answers
106 views

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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1answer
76 views

How to obtain elements of dihedral group in GAP

I'd like to obtain a list of all permutations contained in D_n using GAP. E.g. for D_8 I've tried the following: ...
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1answer
71 views

Semidirect product command in GAP [duplicate]

I need to type the group in GAP. Is it correct if I type as, "SemidirectProduct(ZmodnZ(3),AbelianGroup([7,7]));"? Will it recognize that AbelianGroup([7,7]) is the normal subgroup?
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Obtaining the adjacency matrix of Cayley graphs

Is it possible to obtain the adjacency matrix of a Cayley graph of $Z_3 \times Z_5$? (Manually or by using a software like GAP). Will there be a pattern for adjacency matrices of Cayley graphs for a ...