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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### 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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### 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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### 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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### GAP/HAPcocyclic: How to work with CcGroups

This is probably not a conceptual question- but I would appreciate any suggestions.
I am trying to construct central extensions of certain infinite groups (for example, a crystallographic space ...

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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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### Lie Algebra Module Decomposition in GAP

Let $\mathfrak{g}$ be a complex finite-dimensional Lie algebra and let $V$ be a finite-dimensional $\mathfrak{g}$-module. Is there a way for me to check in GAP or some other software package whether $...

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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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### 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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### 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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### 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-...

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### GAP: Find the decomposition of a (bi)graded module given by generators and relations

I am looking for a way to get the decomposition into indecomposables of a bi-graded modules $M$ over $S = Q[x, y]$ given by it's presentation : a $k \times n$ matrix ...

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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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### 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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### Finding all submodules

Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...

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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 \...