Questions tagged [group-theory]
Questions on the group-theoretic functionality of Mathematica.
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Elements of a group that send one element to another
If I have a permutation group, say $S_{10}$, how do I get all the permutations that send the set {1, 2, 3} to {5, 6, 7}? I know ...
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How to list all subgroups of symmetry group S_6?
I learned from https://oeis.org/A005432 that $S_6$ has $1455$ subgroups, how can I list them all in mathematica?
As the comment says, the direct approach cannot solve the problem.
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How to find a set of generators with the smallest number of elements in a permutation group?
Given a permutation group with many generators
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How to generate all permutation matrices for 4 qubits?
In quantum mechanics, a qubit can be understood as a 2 by 1 vector denoted by Dirac notation as $$|0\rangle \equiv \left( \begin{array}{c}
1\\
0\\
\end{array} \right) ,|1\rangle \equiv \left( \...
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Why am I having issues working with large group multiplication tables?
My mathematica running on Wolfram Cloud seems to break down if I use a SymmetricGroup greater than 6. As a basic example, when I run,
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Cayley for SL group
In this paper, they are using an expander graph. It seems like it's just a Cayley graph for $SL(2,Z_p)$, where $P$ is a prime number.
How do I go about making a Cayley graph as shown in the first of ...
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How to calculate Intermediate fields of Galois group in Hasse diagram?
I can easily calculate the Hasse diagram of an equation in maple, for example for the equation $x^4 + 8x + 12$(It's Galois group is $A4$):
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How to convert a polynomial into monic form of a polynomial
The function ResourceFunction["StauduharGaloisGroup"] can get a Galois Group about a monic irreducible integer polynomial. But I want to know the Galois ...
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How to find all the vertices of graph that have the same status as a vertex $v$?
We sometimes say that in a graph $G$ two vertices $a$ and $b$ look the same. In layman's terms, this means that $a$ and $b$ are of same status. Precisely, there exists a auto-isomorphic mapping of $G$...
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Make GroupOrbits recognise same orbit
When going through a list of elements, GroupOrbits usually recognises if the orbit of an element has been calculated already and will skip it in that case to avoid ...
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How to determine if a group H is a normal subgroup of group G?
A subgroup $H$ of the group $G$ is normal group in $G$ if and only if $\displaystyle ghg^{-1}\in H$ for all $\displaystyle g\in G$ and $\displaystyle h\in H$. How to use MMA to know the group $H$ is a ...
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Finding double cosets of a group
I am new to Mathematica, and I would love to know if there is a function that that returns the double cosets of a group $G$ w.r.t a subgroup $H$, where a double coset is defined as $HxH=\{h_1xh_2,\...
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How to convert a PermutationGroup to a named group
We can convert any built-in named group into PermutationGroup by this code(such as AlternatingGroup[5]):
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How to know the Galois Group of a polynomial is a solvable group?
ResourceFunction["StauduharGaloisGroup"][2 x^5+3 x^4+10 x^3+15 x^2+8 x+12,x]["GaloisGroup"]
...
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How to draw a cycle graph of a group?
MMA can plot a Cayley graph by CayleyGraph directly, which can help us to visualize the group:
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How to get the action diagram when the group act on a set?
I previously thought the orbital map would achieve this, but as the current answer or discuss in the comment. I realized that was wrong. And the Close behavior advice tells me "Needs details or ...
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How to find all left cosets about a subgroup?
I have a group:
group = PermutationGroup[{Cycles[{{1, 3, 6}, {2, 4}}], Cycles[{{6, 7}}]}];
And I have s subgroup
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Source files for the book Exploring Abstract Algebra with Mathematica
I have been looking for the latest version from 2020.
The way back machine did not save the file.
The files that I’m looking for are :
AbstractAlgebraDownloadsV9x.zip
And Master.m
Here is the link ...
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Are there irreducible representations of the crystallographic groups
This yields the list of the irreducible representations of the point crystallographic group Oh:
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Issues in FiniteGroupData Character Tables
Using $Version == "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" and executing
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Replacing multiple variables according to the index
I'm currently working on the Polya's Enumeration Theorem implementation in Mathematica. As an example, of what I want to do, here's a formula I'm working with:
$$P_{C_{12}}(x_1,\ldots,x_{12})=\frac{1}{...
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Reducing to Irreducible Representations
Group Theory Background / Utilities
Suppose I give you a list G of matrices which represent some group, in that the matrices are closed under multiplication. In ...
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CharacterTable for symmetric groups $S_n$ with large $n$
I am looking for a package to generate character tables for symmetric groups $S_n$. At this moment I am using
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Solving abstract factorization problem with Mathematica
I'm a Mathematica newbie.
Given something like:
$$(r_0 + r_1 a + r_2 a^2 + r_3 a^5)(r'_0 + r'_1b + r'_2 b^2 + r'_3 b^5) = 1 - a^5 b^5$$
I'd like to determine if there exist $r_i,r'_i$ that solve the ...
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Packages for study presentation of groups
This question asks if there are tools/packages in Mathematica for the study of presentation of groups, but it is almost 7 years old and an answer suggests to use Combinatorica package which is now ...
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How Can I Compute The Automorphism Group of a Matroid?
I want to compute the automorphism group of a matroid. This reduces to the following (more general) problem:
Suppose I have a list of sets $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ ...
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How to decompose a 5-cycles into a permutationproduct of two 3-cycles?
We know that every 3-cycles can be expressed as the product of two commutations.
Cycles[{{1, 2, 3}}] ==
PermutationProduct[Cycles[{{1, 3}}], Cycles[{{3, 2}}]]
In ...
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How to delete duplicate graphics of the same kind?
A054247: Number of n X n binary matrices under action of dihedral group of the square D_4.
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How to correctly enumerate all the schemes of this cube coloring problem?
This problem is the fifth question of 1996 Chinese High School Mathematics League or Chinese Mathematical Olympiad in Senior:
Choose several colors from the given six different colors to dye six faces ...
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How to create a set (of matrices) which will be used as an finite group to minimize a function?
I'm new here so I'm a little lost.
I need to minimize a function considering that the minimizing parameter belongs to a preestablished set. It all involves matrices.It's something like this:
Where ...
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Orbits of a set $X$ under the action of cyclic permutation $T$
Let $X$ be a set defined as
$$X = \{\{\sigma_1, \dots, \sigma_L\} \;|\; \sigma_i = 0,\dots ,n-1\}.$$ Furthermore, let $T:X\longrightarrow X$ be a cyclic permutation
$$ T\cdot\{\sigma_1, \dots, \...
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Finding invariant matrix given group elements
I have a question about speeding up / optimising the following calculation. I have a feeling there is a way to rewrite it but I can't quite see what to do.
In $d=3$ dimensions, given a set of ...
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Does Mathematica limit the size of a set of permutation group generators?
I tried to generate general and special linear groups PGL2 and PSL2 over finite fields using straightforward algorithms over finite fields: The generators are permutations defined as the additive ...
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Finding sequence of group generators that yields group element
Here is my permutation group that acts on lists of length 4, defined in terms of four generators:
...
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Symmetry unique atom coordinates
One thing I love about Mathematica is how easily I can go from the name of a molecule to estimated coordinates of its atoms, with a command like
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How to compute with Mathematica the space representation of a group for a given point set?
A set of points $P = \{p_1,p_2,\dots\} \subset \mathbb{R}^3$ in 3D is given. A symmetry transformation $S \in \mathbb{R}^{3 \times 3}$ is considered here as a matrix which maps the elements of $P$ to ...
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How to solve problem 599 of Project-Euler
Problem 599 of the Project-Euler is described as follows:
The well-known Rubik's Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible ...
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Calculating Rubiks $ 2 \times 2 \times 2 $ Permutation using Cycles
The help page of PermutationGroup shows a neat example on calculating the permutations of a $3\times 3\times 3$:
...
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Illustration of Motzkin numbers: how to delete duplicates
The $n$-th Motzkin number is the number of different ways of drawing non-intersecting chords between $n$ points on a circle (not necessarily touching every point by a chord -- see https://en.wikipedia....
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How to use Mathematica to prove that isotropic materials have only two independent parameters
Posts on related issues can be found from here or here.
Index symmetries:
A stiffness tensor $C$ is a fourth-order tensor with components $c_{ijkl}$ which maps symmetric second-order tensors into ...
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Delete duplicates when cycle both position and element
Cycle of position means: {a, b, a, a} and {a, a, b, a} is the same.
Cycle of element means: ...
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How to evaluate an expression [closed]
I have a 3-dimensional array of numbers say A. I want to evaluate the following expression.
$\displaystyle\sum_{\sigma,\tau \in S_6}(\Pi_{i=1}^6A(i,\sigma(i),\tau(i)))$. $S_6$ is the symmetric group ...
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What is the cycle index polynomial for the trivial group
CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}]
returns 1.
I was expecting ...
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How to remove global phase from matrices
I have a list of 3 by 3 complex matrices. I would like to remove the matrices that are similar up to a global phase factor.
For instance, I have $ M, \mathrm e^{\mathrm i2\pi/3}M $ and $ \mathrm e^{-\...
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How to list all the possible functions of given structure?
I can run this command :
Tuples[{ Tuples[{x, y, z}, 2], {a, b, c}}]
to list all combinations that looks like:
...
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$2+2+\dots+2$ cycle conjugation classes of the symmetric group $S_n$
Command GroupElements[SymmetricGroup[4]] gives me all cycles of all conjugation classes. But I'm only interested in the 2+2 class ($2+2$ being one of the integer ...
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