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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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How to check whether a finite $p$-group is regular in GAP?

I am trying to check whether a given $p$-group is a regular $p$-group in GAP. I am trying to use the command 'IsRegularPGroup(G)' for it. However I am getting 'Error, Variable: 'IsRegularPGroup' must ...
cryptomaniac's user avatar
-3 votes
0 answers
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To find exponent of $1+J(F_2G)$ in GAP. [closed]

Please provide code in GAP to find the exponent of the normal subgroup $1+J(F_2G)$ of the unit group $U(FG)$ of the group algebra $FG$. Here $J$ stand for the Jacobson radical. Write the expression by ...
neelkanth's user avatar
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1 vote
0 answers
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IsGroupOfAutomorphisms functionality

I'm looking for two things related to the GAP function IsGroupOfAutomorphisms: whether it does what I think it does (based on the brief GAP manual entry), and if so, how it works. The GAP manual ...
Michael Wynne's user avatar
0 votes
1 answer
76 views

Computing whether two finite groups are isomorphic (in C++) [closed]

I need to algorithmically compute whether two given finite groups are isomorphic. Usually I only have generators of these groups. The groups can get quite large as I'm working with subgroups of $S_{32}...
H-a-y-K's user avatar
  • 729
-1 votes
1 answer
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Character table for a covering group of $\mathbb{Z}_n \times \mathbb{Z}_m$

I’m considering the group $G = \mathbb{Z}_n \times \mathbb{Z}_m$ and its covering group $$G^* = \langle \alpha, \beta, a|\alpha a = a\alpha, \beta a = a\beta, a^p = 1, \alpha^n = 1, \beta^m = 1, \...
slowspider's user avatar
  • 1,065
5 votes
1 answer
72 views

Character tables of Coxeter Groups

I'm interested in character tables of (irreducible) Coxeter groups. Certainly the character tables of the symmetric groups $ W(A_n) \cong S_{n+1} $ are easy to obtain, as are the dihedral groups. But ...
Ian Gershon Teixeira's user avatar
1 vote
2 answers
76 views

Counting orbits of permutation group acting on bit strings

Let $ G $ be a subgroup of $ S_n $. What is the the best way to count the number of orbits of $ G $ acting on the length $ n $ bit strings $ \mathbb{F}_2^n $? Obviously permutations can only take a ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
78 views

Multiplicity of irreducible characters inside symmetric powers of a faithful character

Let $ G $ be a finite group of order $ n $. Let $ f $ be a faithful representation of $ G $ and by abuse of notation let $ f $ also denote the corresponding character. It is my experience that for ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
84 views

Automorphism group of $A_n$, $n \geq 7$ [duplicate]

I am trying to find the automorphism group of the alternating groups $A_n$. However, when it comes to $A_7$, I have found it difficult to prove that $\operatorname {Aut}(A_7) \cong S_7$. (I have ...
tys's user avatar
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Restriction of Brauer characters in GAP

Let $p$ be a prime. Let $G$ be a finite group and $H$ be a $p'$-subgroup of $G$. Let $\varphi\in \mathrm{IBr}_p(G)$. I would like to restrict $\varphi$ to $H$ by GAP. My idea is to extend $\varphi$ to ...
user44312's user avatar
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2 votes
1 answer
37 views

GAP orthogonal groups: Specifying the invariant bilinear form

If I read the GAP manual correctly, one should be able to specify the underlying invariant bilinear form, when constructing an orthogonal group. However, when I try something like: ...
Fungaria's user avatar
3 votes
0 answers
50 views

Burnside groups with GAP system [closed]

My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP. The obvious representation using relations (see example for ...
arthurbesse's user avatar
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Factoring out the socle of the projective-injectives for a quiver algebra

Let $A$ be a quiver given in the GAP-package QPA (https://folk.ntnu.no/oyvinso/QPA/) . Question: Is there a fast/easy way to obtain $A/soc(U)$ (using QPA), where $U$ is the direct sum of all ...
Mare's user avatar
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1 vote
1 answer
46 views

Ordering of group elements in MultiplicationTable vs CosetTable in GAP

I construct a finitely presented discrete group $G$ in GAP, a normal subgroup $H\triangleleft G$ of finite index $N$, and the factor group $G/H$ of order $N$. Assume for simplicity that $G$ has two ...
MathPhysGeek's user avatar
1 vote
0 answers
58 views

Checking if matrices in $ SU(2) $ generate an $ S $-arithmetic group

I am reading Super-Golden-Gates for $PU(2)$ by Ori Parzanchevski and Peter Sarnak. If I understand correctly then in section 4.1.3 they seem to be saying that the matrices $$ F:= \frac{1}{\sqrt{2}} \...
Ian Gershon Teixeira's user avatar
0 votes
1 answer
69 views

how to calculate the automorphisms of a group that fix a subgroup

I have a finite (polycyclic) group $G$ and a subgroup $H<G$. How do I calculate the subgroup of $Aut(G)$ that fixes $H$ pointwise : $S = \{ a \in Aut(G) : \forall h \in H,a(h)=h\}$ It would be nice ...
unknown's user avatar
  • 1,010
6 votes
1 answer
77 views

Can GAP compute this 16-dimensional representation of AlternatingGroup(6)?

I am interested in a particular 16-dimensional representation of $A6$, the alternating group on 6 things. I first construct an amalgam, gamma, of two copies of SymmetricGroup(4): ...
Jesper M. Moller's user avatar
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0 answers
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About a presentation of a FpGroup

I'm studying a group in GAP, and I have a list of relators of this finited presented group. My trouble is, this group has too many relations (actually more than 500), and GAP gives me these relations (...
none's user avatar
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2 votes
0 answers
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Are all automorphisms of the character table of a quasisimple finite group generated by Galois and outer automorphisms?

This is a follow up question to Automorphisms of a character table Consider the character table of a finite group $ G $. Every outer automorphism of $ G $ permutes the characters of $ G $. If some of ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
79 views

Find a relation in GAP

I have a finitely presented group $G$ and a finitely generated subgroup $H<G$. GAP computed that $H$ has finite index in $G$. However, PresentationSubgroupMtc(G,H) cannot compute the presentation ($...
QMath's user avatar
  • 427
2 votes
1 answer
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Automorphisms of a character table

Consider the character table of the finite group $ G $. Every outer automorphism of $ G $ permutes the characters of $ G $. If some of the characters are non-integral then we can consider the ...
Ian Gershon Teixeira's user avatar
8 votes
1 answer
115 views

Distinct characters with the same character values, outer automorphisms and Galois conjugation

Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree: multiplying by a degree 1 character applying an ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
78 views

Finding orbits of an action on a set of basis vector with GAP

I work with the computer algebra system GAP in this question. Let $K$ be a field (for example the rational numbers). I have a set $W$ consisting of sets of vectors basis of $K^n$ that only have ...
Mare's user avatar
  • 2,332
0 votes
0 answers
55 views

Intersection of Ideal in Gap(QPA)

I'm New to Gap(QPA). I'm trying to intersect Ideal in Gap and find their generators. For example here I construct some trivially intersecting ideals, Q:= Quiver(1,[[1,1,"a"],[1,1,"b&...
Mukilraj K's user avatar
1 vote
1 answer
91 views

Create matrix from permutations

I have a set of four permutations $\{(\cdots),(3,4),(2,4),(2,3)\}$. I want to create a $4 \times 4 $ matrix with rows as above permutations. For example in this case matrix will be \begin{bmatrix} 1 &...
zigzag's user avatar
  • 35
6 votes
1 answer
154 views

How often is a tensor product of two irreps of a finite group still irreducible?

All representations are considered over the complex numbers. Let $ G $ be a finite group. Then Any 1-dimensional character $\otimes$ irreducible character is irreducible . But what if we have two ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
86 views

Comparing $ GL(2,3) $ and the binary octahedral group

$ \mathrm{GL}(2,3) $ is the group of $ 2 \times 2 $ matrices over the field with 3 elements. It has $ 48 $ elements and it is the Schur cover of $ S_4 $ of $ + $ type, fitting into a central extension ...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
128 views

Estimate the subgroup order of group of units in finite algebra in GAP

Let $G$ be a finite group $p$-group and $FG$ be a modular group algebra of $G$. Is there any way to estimate the order of subgroup generated by two or more elements of $U(FG)$, where $U(FG)$ is the ...
limakzi's user avatar
  • 227
0 votes
1 answer
50 views

Counting groups based on PSU(4,3)

How many almost quasisimple extensions of PSU(4,3) are there? As noted in Constructing central extensions or Schur cover of U4(3) in GAP this group is too large to be in the perfect groups library (...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
33 views

N-dimensional linear representation of symmetric group S(N)

A beginner's question on the linear representations of the symmetric group $S(n)$: playing around with GAP, e.g.: gap> CharacterDegrees(SymmetricGroup(n)); for ...
user71769's user avatar
  • 354
0 votes
1 answer
54 views

Is the affine special linear group perfect?

The special linear group $ SL(n,q) $ is perfect, in fact quasisimple, for all $ n \geq 2 $ and $ q $ prime power with the exception of $ SL(2,2) \cong S_3 $ and $ SL(2,3)\cong 2.A_4 $. Is it the case ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
206 views

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, ...
Aiden Peterson's user avatar
3 votes
2 answers
102 views

Free Group where generators aren't just one symbol?

The question is: Show that $G = \langle a, b, c \mid a^2bacacab \rangle$ is a free group on the free generators $ab$ and $ac$. This is in section 1.4, page 39 of Combinatorial Group Theory by Magnus, ...
N A's user avatar
  • 51
1 vote
0 answers
37 views

Lie algebra of the group generators in GAP [closed]

I have generators for my group in GAP. Additionally, I possess a specific unitary element and wish to determine whether this unitary element arises from the Lie algebra of my group's generators. ...
j.doe's user avatar
  • 217
1 vote
1 answer
173 views

How to construct different modules using Gap?

I want to calculate the cohomology of some space group using Gap, with the coefficient different $\mathbb{Z}$-modules corresponding to different group actions on $\mathbb{Z}$. For example, for the ...
Ye Weicheng's user avatar
1 vote
1 answer
81 views

Finding automorphisms of a vector space using GAP

Goal: Given a $K$-dimensional vector space $V \subseteq \mathbb{F}_2^N$ firstly find the group $A$ of all automorphisms $\alpha_i: V\to V $ and secondly find the subgroup $\widetilde{A} \subseteq A$ ...
Dreieck Dreieck's user avatar
0 votes
0 answers
34 views

Can you solve $X A X^T = B$ for $X$ over Gaussian rationals in GAP?

I have two symmetric matrices : $A$ is $a \times a$ and $B$ is $b \times b$. I'm trying to find solutions for $X A X^T = B$. $A$ and $B$ are constant and $X$ is the unknown $b \times a$ matrix. All ...
unknown's user avatar
  • 1,010
3 votes
1 answer
87 views

Constructing central extensions or Schur cover of U4(3) in GAP

Part of my group theory project involves looking at the group $U_4(3)$, which has abnormally large Schur Multiplier (36) and large automorphism group ($D_{12}$). I need to work with the central ...
abiteofdata's user avatar
0 votes
1 answer
43 views

Converting a group element to an ordered list of generators in GAP

Consider the free group on {a,b} which is gap is constructed with FreeGroup("a","b"). I want to cyclically reduce words. For example, given word $a^{-1}*b*a$, this would be ...
Mithrandir's user avatar
-2 votes
1 answer
92 views

Isomorphism and GAP [closed]

Suppose $G$ and $H$ are two finite groups such that $\frac{G}{Z(G)}\cong\frac{H}{Z(H)}$. The Command $K:=IsomorphismGroups(\frac{G}{Z(G)},\frac{H}{Z(H)})$, in GAP, gives all the isomorphism from $\...
D. N.'s user avatar
  • 2,221
0 votes
1 answer
48 views

Viewing monoid rings as rings with identity in GAP

I look at monoid algebra of finite monoids with GAP and want to force GAP to view them as algebras with one. But it seems it does not work: ...
Mare's user avatar
  • 2,332
4 votes
1 answer
421 views

All groups of order $112$ have an element of order $14$ [closed]

In finite groups there exists a very important class as Frobenius groups. We know that there exists a Frobenius group as $2^3:7$ which has an elementary abelian $2$-group. By the Properties of ...
Brz K74's user avatar
  • 75
0 votes
1 answer
64 views

Choosing Particular representation and/or identifying the representation of the group in GAP [closed]

I have three questions regarding the usage of GAP: I want to choose a particular matrix representation of a group, and thanks to the previous answer, I can display which representations are available,...
j.doe's user avatar
  • 217
1 vote
1 answer
122 views

How to identify a group as a primitive group?

I have a problem identifying a group as a primitive group by using the function PrimitiveIdentification. Here is my code. ...
user44312's user avatar
  • 521
1 vote
1 answer
91 views

Does every perfect finite group have a fixed point free representation?

Fixed point free representations of finite group are important for the spherical space form problem and also show up in other contexts for example Perfect semi direct products A representation $ \pi $ ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
26 views

Writing the group over cyclotomic fields in GAP [duplicate]

I am able to define the group and do something with this group in GAP. For example, I am interested in the 2I group, and I can write this group in GAP simply as follows: ...
j.doe's user avatar
  • 217
2 votes
0 answers
131 views

Is it possible to write a code that computes all the subgroups (up to isomorphism) of a finite Abelian group?

I am new to computational group theory. I am trying to find/write a program that computes the following: Input: a finite abelian group $$G \cong \mathbb{Z}_{m_1} \oplus \cdots \oplus \mathbb{Z}_{m_{...
ghc1997's user avatar
  • 1,641
2 votes
1 answer
86 views

How to make characteristics zero in GAP?

I have the following code written in GAP. In summary, In this code, I have a subgroup defined as SL(2,5). I applied the tensor product to each of these group elements with the identity and added a new ...
j.doe's user avatar
  • 217
1 vote
1 answer
85 views

Canonical representation for sets in a group.

Let $G$ be a permutation group acting on a set $X$ of elements. Is there a known group theory operation "$\mathrm{Canon}$" such that, for each pair $A, B \subseteq X$ of elements subsets, $\...
FxMySz's user avatar
  • 45
1 vote
0 answers
94 views

Finding and Identifying Finite Subgroups in SageMath

I have a finite subgroup, which I determined to be finite using the is_finite() function. The cardinality() function helped me ...
j.doe's user avatar
  • 217

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