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.
74 questions
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Program for computing group cohomology
Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space?
I mainly care about infinite groups.
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Algebra for the Baby
I am reading the following article.
Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012..
Author works with 4370-...
user21230
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On the sum of the subgroup orders of a finite group
Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum_{H \le G} |H|.$$
Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
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Non-Boolean Eulerian interval of finite groups
An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It ...
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Quiver and relations for blocks of category $\mathcal{O}$
In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ .
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Computing homology groups with GAP
I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
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Irreducible tensor product representations in finite simple groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
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Algorithm for finding quiver algebras
Im looking for an algorithm that does the following in a quick way:
Input: Natural number $r \geq 2$, natural number $s \geq 3$, prime power $q$.
Output:
Finds all two-sided ideals in $J^2/J^s \...
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Lower bound for the order of a simple group with a given class number
Every simple group below are assumed non-abelian.
Let us call the class number $k(G)$ of a finite group $G$ the number of its conjugacy classes (also, the number of its irreducible complex ...
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Maximal numbers of summands in middle terms of short exact sequences
Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split ...
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Membership to double cosets in free groups
Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...
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recognition of symmetric groups in GAP
In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes ...
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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 a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example,
\begin{align}
n^+ =
\begin{pmatrix}
...
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Are these two kernels isomorphic groups?
We have a finitely presented, infinite group $\mathsf{B}$,
coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface).
It is generated by elements
\begin{...
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Is It possible to determine whether the given finitely presented group is residually finite with MAGMA or GAP?
I am working on finitely presented groups with more than 5 generators and relators and I'm so curious: is it possible to determine residually finitness of finitely presented groups with MAGMA or GAP?
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Is there a maximal subgroup of depth 3?
Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
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Get the commands history from GAP system
I am not sure whether this was asked before, but I didn't find a reference in GAP system documentation on how to print the history of the command line (Ubuntu installation).
For instance:
...
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A question about the possibilities of GAP
Let $R=\mathbb{Z}/1024\mathbb{Z}$ and $G=GL(3,R)$. Let $H$ be the subgroup of $G$ consisting of all matrices with determinant $1$ which are congruent to the identity matrix modulo the ideal $4R$. Let
$...
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Obtaining quiver and relations for finite p-groups
Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)?
Since $KG$ is local, the quiver should ...
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Is there a subgroup of dual depth 3?
This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end...
Let's ...
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Conjugacy classes in normalized unit group of a group ring
Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
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A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
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normal form for some finite groups, extending the small groups library
I am in need of a normal (that is, canonical) form for (some) finite groups, computable with - for example - gap or sage or any other freely available package. The goal is to make finite groups ...
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Database subgroups of free group
Is there some database that contains "all" low-index normal subgroups of the free group on two generators?
Extension: does there exist such a GAP-database?
Thank you!
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The associated graded algebra of a finite dimensional algebra
$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).
Denote by $A_G$ the associated ...
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GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...
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Testing whether a module generates $K_0(\mbox{mod-}A)$
Given a representation-finite (connected) quiver algebra $A$ and a module $M$.
Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$?
Can ...
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A global code for the character table of PSL(2,q)
We can easily get the character table of $\mathrm{PSL}(2,q)$ for some fixed small prime power $q$, we can just do (for example):
...
4
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Where or how can I find matrix representatives of the conjugacy classes of Conway's group Co₀?
I would like to find ($24\times 24$) matrices representing the various conjugacy classes of Conway's group $\mathrm{Co}_0$ acting on the Leech lattice in the usual coordinate system given by the MOG. ...
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Finding all nice ideals for quiver algebras
Let $Q$ be a finite, connected and acyclic quiver which is simply-laced.
Let $k$ be a field and $kQ$ the path algebra of $Q$ over $k$.
Recall that an ideal $I$ of $kQ$ is called admissible if it is ...
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Indexed character tables for wreath products in Sage and GAP
I am trying to obtain character table for the Hyperoctahedral group $\mathcal{H}_n$ in Sage using GAP.
This group arises as the wreath product $\mathcal{C}_2 \wr \mathcal{S}_n$, so of course I can ...
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Finding local algebra and relations lottery
This can be seen as an attempt for a mini Polymath project on homological properties of (local) finite dimensional algebras. You only need to know what a finite dimensional algebra is and have GAP to ...
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Commutative algebras associated to simple Lie algebras
In Section 2 of the article https://www.sciencedirect.com/science/article/pii/S0021869307000385, the authors study the center $Z=Z_Q$ of certain preprojective like algebras associated to the simply ...
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Field elements in quiver and relations
Let $A=KQ/I$ be a quiver algebra such that the coefficients of the relations in the admissible ideal $I$ consist only of the field elements $0,1$ and $-1$.
Question 1: Is it true for every basic ...
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Recovering the bimodule from the trivial extension
Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...
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Interpreting $H^n(BG,\mathbb Z)$ when $G$ is an infinite discrete group
Suppose $G$ is a two-dimensional space group, for example a semidirect product of $\mathbb Z^2$ with a crystallographic point group such as $\mathbb Z_2$, where the action of $\mathbb Z_2$ on $\mathbb ...
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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 ...
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Frobenius-Schur indicator and character table of finite groups
Let $G$ be a finite group and $\pi$ an irreducible complex representation. The Frobenius-Schur indicator of $\pi$ is defined as:
$$ \nu_2(\pi):=\frac{1}{|G|} \sum_{g \in G} \chi_{\pi}(g^2) $$
with $\...
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Finding the non-trivial block of a finite dimensional algebra via GAP
Let $A$ be a finite dimensional $K$-algebra that has a block decomposition $A=A_1 \times A_2 \times \dots\times A_n$. (we can assume that $A$ is a quiver algebra if that helps, meaning all simple $A$-...
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Calculating cohomology group $H^3(point group,\mathbb{Z})$ using GAP program
I'm trying to compute $H^3(point group,\mathbb{Z})$ for all the 32 point groups in 3D which has some applications in physics. Unfortunately, I could not find literature discussing this problem. So I ...
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Modules with special properties
$\DeclareMathOperator\End{End}$Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $\End_A(M)$ annihilates the socle $\...
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Thin representations for quiver algebras
A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries.
When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that ...
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Finding automorphisms and cyclic modules via QPA
Given a symmetric finite dimensional algebra $A$ over a finite field with enveloping algebra $A$.
Assume we know that $\Omega_{A^e}^i(A) \cong A_{f}$, where $f$ is some automorphism of the algebra $A$....
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Isomorphism and counting for tree quivers
Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
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Finding a bigger Frobenius algebra for a given local algebra
Let $A=K\langle x_1,\ldots,x_n\rangle/I$ be a local finite dimensional algebra with admissible relations $I$.
Question: Is there a canonical way to check whether $A$ is isomorphic to $B/\operatorname{...
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Quiver algebras of Dynkin type
Let $kQ$ be one of the Dynin path algebras of type $A_n , D_n $ or $E_i$ for $i=6,7,8$.
Question 1: How many (up to isomorphism) quiver algebras are there that are derived equivalent to $kQ$?
...
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Differences between GAP and MAGMA [closed]
GAP and MAGMA are computer algebra systems. What are the objective differences between the two?
Which capabilities are not shared?
How do they compare on facilities for working with character tables?...
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Use GAP program to obtain explicit cocycles in group cohomology
I'm trying to compute group cohomology $H^n(G,\mathbb{Z})$ of some crystal groups $G$ which are infinite but finitely generated groups. I succeed in obtaining cohomology groups using projective ...
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Union of the conjugates of maximal subgroups
This post is a generalization of Union of the conjugates of a proper subgroup.
Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that:
(1) $ \...
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Preprojective algebra of finite dimensional algebras
The preprojective algebra of a module $M$ over a finite dimensional algebra $A$ is
defined as $P_M:= \bigoplus\limits_{n=0}^{\infty}{Hom_A(M, \tau^{-n}(M))}$ with the canonical multiplication.
...
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