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
799
questions
0
votes
1
answer
38
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 ...
- 7,198
-1
votes
0
answers
47
views
GAP Isomorphism between two finitely presented p-groups [closed]
I have finitely presented p-groups G and H of order 5^6 and i want to find isomorphism between them. I need to know image of generators [x1, x2, x3, x4] of group G. I tried using ...
-4
votes
0
answers
34
views
Problem in using LogTo in gap [closed]
I am trying to use
$LogTo("test.log")
command to save my files but getting following error
...
3
votes
2
answers
95
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, ...
- 51
1
vote
0
answers
30
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. ...
- 217
0
votes
0
answers
32
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 ...
1
vote
1
answer
55
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 $\widetilde{A} \subseteq A$ ...
0
votes
0
answers
32
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 ...
- 958
2
votes
0
answers
38
views
Constructing central extensions or Schur cover of U4(3) in GAP
Part of my group theory project involves we're 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 ...
- 105
0
votes
1
answer
38
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 ...
- 784
-2
votes
1
answer
83
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 $\...
- 2,211
0
votes
1
answer
45
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:
...
- 2,300
4
votes
1
answer
394
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 ...
- 75
0
votes
1
answer
62
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,...
- 217
1
vote
1
answer
74
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.
...
- 503
1
vote
1
answer
60
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 $ ...
- 7,198
0
votes
0
answers
25
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:
...
- 217
2
votes
0
answers
122
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_{...
- 1,277
2
votes
1
answer
81
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 ...
- 217
1
vote
1
answer
75
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, $\...
- 43
1
vote
0
answers
59
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 ...
- 217
3
votes
0
answers
82
views
Infinite Groups in SageMath(or in GAP)
I am once again asking more software-related questions, but I am happy to learn a general answer as well. I'm curious about how the is_infinite function works in SageMath (or in GAP). Does it simply ...
- 217
1
vote
1
answer
72
views
Understanding whether the addition of an element to a subgroup of an infinite group will result in an infinite group or not in GAP/SageMath
This is more of a software-related question, but I wanted to ask here because I am confident that most people in this forum are well-acquainted with GAP/SageMath and can provide more insight for the ...
- 217
0
votes
0
answers
53
views
Representatives of conjugacy classes of GL(n, p)
I am interested in using GAP to compute conjugacy classes of subgroups of the general linear group of $n \times n$ matrices over $\mathbb{F}_p$, denoted as $\text{GL}(n, p)$. In a previous question it ...
- 21
1
vote
0
answers
33
views
Obtaining split extensions with GAP
Let $A$ be a finite dimensional $K$-algebra (K a field) and $I$ a two-sided ideal of $A$.
(if it helps we can assume that $A$ is a connected quiver algebra or a group algebra and $I$ a power of the ...
- 2,300
0
votes
1
answer
49
views
Group action on direct product in GAP
I want to define the group action of $G$ on $G \times G$ by left multiplication. For example I have G:=DihedralGroup(6);. I created ...
- 23
0
votes
0
answers
66
views
Computing irreducible representations of larger groups with GAP
I'm currently trying to use GAP's function IrreducibleRepresentations() to compute the complex irreducible representations of some group $G$. We're using this function to determine whether each ...
- 105
0
votes
0
answers
44
views
Calculation in a Group Algebra Using GAP.
I am having no knowledge of GAP but I want some calculations in group algebra $FG=F_pD_{2n}$. Here $F_p$ is a finite field with $p$ elements and $D_{2n}=\{a, b\mid a^n, b^2, (ab)^2=1\}$ is Dihedral ...
- 171
4
votes
1
answer
151
views
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
2
votes
1
answer
133
views
GAP: How to find a (possibly non-minimal) transitive representation of a group?
I've recently fell in love with Galois Theory again and in an attempt to categorize how one could determine the Galois group of small-degree irreducible polynomials, I stumbled upon this small problem ...
- 846
3
votes
1
answer
97
views
Non-split extension of an extraspecial 3-group by SL(2,3)
Let $Q$ be an extraspecial $3$-group of order $3^5$. I am curious about the existence of a group $G$ satisfies the following condition.
(1) $G$ is a non-split extension of $Q$ by the special linear ...
- 503
2
votes
1
answer
97
views
Characters of almost quasisimple groups
A group $ Q $ is called quasisimple if it is a perfect central extension of a simple group.
A group $ A $ is called almost simple if there exists a series of containments $ S \subset A \subset Aut(S) $...
- 7,198
0
votes
1
answer
68
views
Testing if an element is in the derived subgroup
Let $ G $ be a finite group. Let $ G':=[G,G] $ be the derived subgroup. How do you test if an element $ g \in G $ is in $ G' $?
I'm specifically interested in how to do this in GAP. In other words I ...
- 7,198
1
vote
0
answers
74
views
Maximal quotient group of direct product
In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$
We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such ...
- 53
2
votes
1
answer
72
views
Finitely presented finite subgroups of a finitely presented infinite group in GAP?
I am trying to work with infinite Coxeter groups in GAP. An example of such a group would be $\langle a, b, c | a^2, b^2, c^2, (ab)^4, (ac)^2, (bc)^4\rangle$ which can be used to describe the 2D (...
- 21
0
votes
0
answers
51
views
On certain abelian subgroups of $S_{12}$ and $S_{18}$
Let $p$ be a prime, and $\Sigma\le S_{p-1}$ an abelian subgroup of order $p-1$ such that:
all the elements of $\Sigma$ have cycle type $\underbrace{\left(\frac{p-1}{k},\dots,\frac{p-1}{k}\right)}_{k\...
- 2,617
3
votes
0
answers
52
views
Finding a splitting field for a semisimple algebra in GAP
Question: Given a semisimple finite dimensional algebra over the rationals in GAP.
Is there a way to find a (minimal) splitting field of this algebra with GAP and an explicit Wedderburn decomposition?
...
- 2,300
1
vote
1
answer
58
views
DirectSumMat for general fields in GAP
The GAP-command DirectSumMat allows to take the direct sum of matrices over (only certain?) fields.
Now I did the following in GAP:
...
- 2,300
0
votes
0
answers
46
views
Structure of the Jacobson radical of the Group Algebra .
Is it possible to find the structure description of the Jacobson radical $J(FG)$ of a group algebra FG, where F and G are finite field and group respectively in GAP?
I choose the group algebra $F_3D_{...
- 5,932
2
votes
0
answers
36
views
Exterior algebra of a vector space in GAP
In https://docs.gap-system.org/doc/ref/chap61.html section 61.13-2 ExteriorPower one can find how to construct the exterior power of a vector space (lets say $K^n$ where $K$ are the rationals).
Here ...
- 2,300
0
votes
1
answer
76
views
The normalized unit group using GAP.
I want the structure of The normalized unit group using GAP for the group algebra $FD_{30}$, where $F$ is a finite field with characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
- 5,932
1
vote
0
answers
43
views
Unit group structure GAP code. [duplicate]
I want the structure of the unit group of the group algebra $F_{3^k}D_{30}$ using GAP, where $F_{3^k}$ is any finite field of characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
- 5,932
1
vote
0
answers
36
views
Proof that distinct irreps have disjoint projectors
Let $ \chi $ be the character of an irrep of a finite group $ G $. Let $ \pi $ be a representation of $ G $. Then the projection onto the $ \chi $ isotypic subrepresentation is given by
$$
\Pi_\chi= \...
- 7,198
1
vote
1
answer
79
views
Irreducible representations of the symmetric group in GAP
I try to find the irreducible representation of the symmetric group (over the complex numbers) in a quick way using GAP.
I made the following program that gives the irreducible representation ...
- 2,300
0
votes
0
answers
35
views
Computing power of element in GAP [duplicate]
I am trying to find powers of element in GAP; I defined a group G;
When I print G, it shows Group( [ a1, a2, a3, a4, a5, a6 ] );
But if I compute a1^3; It shows "Error, Variable: 'a1' must have a ...
- 2,424
1
vote
1
answer
85
views
Group action in GAP
I want to define a group action of group $G$ on set $X$, $G \times X \to X$ in GAP. I read the manual but I couldn't understand how. For example, I want to define group action of group $G$ on itself ...
- 23
-4
votes
1
answer
87
views
Groups, Algorithms and Programming (GAP 4) lack of functions
I am a newbie in using of GAP. To learn it I downloaded a book "Groups, Algorithms and Programming" (subtitle GAP 3.4.4 of 20 Dec. 1995, gap3-jm of 19 Feb. 2018) and also Reference Manual of ...
- 39
0
votes
0
answers
50
views
Irreducible representation over finite field GAP
I have a finite group $G$ with $p\not \mid |G|$ and I want to compute an explicit modular representation of $G$ given a character $\chi$. I can compute a representation by ...
- 545
0
votes
0
answers
30
views
Constructing a semidirect product in GAP using characters
I have a finite group $G$ and an elementary abelian group $E(p^h)$ where $p\not \mid |G|$. I just want to construct via GAP the semidirect product $E(p^h):G$ given by a character $\chi$ of $G$, not ...
- 545
1
vote
1
answer
65
views
Can GAP determine whether a local algebra is Frobenius?
Let $A$ be a local(not necessarily commutative) finite dimensional algebra over a (finite if it helps) field $K$.
Is there a way to check with GAP (without using QPA as we do not know quiver and ...
- 2,300