Questions tagged [group-theory]

Questions on the group-theoretic functionality of Mathematica.

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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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3 votes
3 answers
191 views

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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8 votes
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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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How to hightlight CayleyGraph?

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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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3 votes
1 answer
63 views

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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1 answer
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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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Mathematica gives invalid generators of a group? [closed]

...
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3 answers
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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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4 votes
2 answers
304 views

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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1 answer
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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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3 votes
2 answers
91 views

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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6 votes
1 answer
82 views

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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5 votes
2 answers
200 views

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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1 answer
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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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4 votes
1 answer
124 views

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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6 votes
4 answers
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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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4 votes
2 answers
365 views

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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2 votes
1 answer
89 views

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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2 votes
1 answer
48 views

What is the cycle index polynomial for the trivial group

CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}] returns 1. I was expecting ...
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3 votes
2 answers
281 views

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 votes
1 answer
38 views

$2+2+\dots+2$ 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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1 vote
1 answer
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What is the mathematical operation corresponding to this definition of the Dirichlet Character group operation?

The Wolfram Language & System Documentation Center page for DirichletCharacter indicates Dirichlet characters modulo k form ...
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1 vote
1 answer
42 views

Show result of matrix operation in terms of user defined matrices

I have defined some matrices K[0],K[1],...,K[9] and S[1],...
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3 votes
1 answer
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The product of two symmetric groups acting on a function

Considering the rational function $$ \small \begin{align*} f&(x_1,x_2,x_3;y_1,y_2,y_3)\\ &=\frac{\left(1-\frac{y_1}{x_1}\right)\left(1-\frac{y_2}{x_1}\right)\left(1-\frac{y_3}{x_1}\right)\...
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2 votes
0 answers
46 views

How to iterate over Orbit of PermutationGroup?

Another day, another permutation group question. I'm given a g=PermutationGroup[...], and a list l={1, 3, 3, 2, ...}. I know I ...
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8 votes
1 answer
605 views

Definition of WignerD function?

On Wikipedia, elements of Wigner's D-matrix are defined as $$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-...
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4 votes
2 answers
120 views

The set of polynomials under the action by a symmetric group

Let $$f(x_1,x_2,x_3)=\frac{x_1^r x_2^r \left(1-x_1 x_3\right) \left(1-x_2 x_3\right)}{\left(1-\frac{x_2}{x_1}\right) \left(1-\frac{x_3}{x_1}\right)\left(1-\frac{x_3}{x_2}\right)},$$ where $r$ is a ...
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7 votes
1 answer
315 views

Irreducible representations and conjugacy classes for the octahedral group

We need a few things about the octahedral group (including reflection, perhaps even using double cover later on): Conjugacy classes Irreducible representations (irreps) in the form of matrices. Just ...
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2 votes
1 answer
147 views

How to create a group via group relations?

I have a group described by group relations. For simplicity group has a presentation $<a | a^n = e>$. How can I create this group in Mathematica?
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6 votes
1 answer
162 views

Generate list of tuples, modulo PermutationGroup

I have a permutation group, e.g. g = PermutationGroup[{Cycles[{{1, 2}}]}] but not necessarily limited to a single generating cycle. What I want is to create a ...
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1 vote
1 answer
84 views

Find a matrix $X$ that block-diagonalizes a particular group of matrices

Essentially, I want to find a single matrix $X$ such that conjugation by $X$ sends: $$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 &...
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  • 160
4 votes
1 answer
91 views

What does this graph mean?

Mathematica 11.3.0 includes a new command FindEquationalProof having good prospects. Studying it, I consider a somewhat modified example from the help ...
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3 votes
1 answer
56 views

Not too obsolete package extending the built-in functionality of Permutation Groups

Is there a modern package geared to permutation groups? With functions, for instance, for group-isomorphism (either w.r.t. permutation representation & w.r.t. group structure), direct, semidirect ...
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