# 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.

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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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### Orbits/affine spaces in GAP

Another GAP-related question.
I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...

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### Inertia indices in GAP

Not sure that this is the right place, but I could not find a GAP specific forum.
Does anyone know if there is a built-in function in GAP to find the inertia indices of a symmetric matrix, say, over ...

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### Implementation of the nerve of a category in GAP

I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...

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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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### 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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### Software for Intersection of Ideals in Noncommutative Polynomial algebra

I am looking for software which can compute an intersection of ideals (in particular right ideals) in a noncommutative polynomial algebra and then find its Gröbner Basis.
Most software somehow does ...

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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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### 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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### 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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### Calculating simple and their syzygies in group algebras with GAP

Let $KG$ be a a group algebra for a finite field $K$ and a finite group $G$.
Question 1: How to obtain using GAP a finite field extension $L$ such that $L$ is a splitting field for $KG$ and all the ...

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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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### Rigid modules for hereditary algebras

Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps)
Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...

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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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### 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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### 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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### Counting the number of generating triples of various types in finite simple groups

I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ...

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### How do I find hyperbolic generating triples for a group using GAP?

Let $G$ be a finite group and $x, y, z \in G$. A hyperbolic generating triple for $G$ is a triple $(x, y, z) \in G\times G\times G$ such that
$\frac{1}{o(x)}+\frac{1}{o(y)}+\frac{1}{o(z)} <1$,
$\...

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### Defining the action of a group on a set in GAP [closed]

Let $H$ be a subgroup of the automorphism group of a group $G$ and $S$ is a collection of subsets of the group $G$ with a given size. The group $G$ is defined via free group and relations in GAP. ...

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### When does a stable endomorphism ring have injective dimension at most one?

tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...

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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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### 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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### Recall command in GAP system [closed]

Is there is a way to recall a command from the history on the Ubuntu GAP system prompt? (something like CTL-R on Linux systems). Basically I want to reuse a command that I typed before, but the only ...

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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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### 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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### 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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### 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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### Finite simple groups and negative Frobenius-Schur indicator

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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### 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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### 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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### 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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### Problem while multiplying under a set of relators [closed]

I have defined $S_4$ (Symmetric group of order 4), and with the base field $Z_5$, groupring $Z_5S_4$ is constructed. Then I have taken two elements of this group ring and I want to multiply them to ...

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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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### 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):
...

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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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### 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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### 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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### Cayley graphs on $Z_{11}$ and $Z_p$

I want to find all cayley graphs on $Z_{11}$. I know how many connected cayley graphs exist but i want to find all of them, connected or not, to find their eigenvalues. I found some of them and a ...

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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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### 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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### 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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### Local submodules of finite rings

Let $A$ be a finite local ring $R$, such as a group algebra of $p$-groups over a finite field of characteristic $p$.
Question 1: Is it possible, using GAP, to obtain the poset of all submodules of $R$...

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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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### On monomial and $\Omega^d$-finite algebras

Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra.
It is well known that monomial algebras ...

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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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### 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 ...

2
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0
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### About $E(G)$ for a finite $p$-group $G$

For any group $G$, the absolute center $L(G)$ of $G$ is defined as
$$L(G) = \lbrace g\in G\mid \alpha(g)=g,\forall\alpha\in Aut(G)
\rbrace$$, where $Aut(G)$ denote the group of all automorphisms of
$G$...