Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields, etc.). For questions about generic computer algebra systems, use [tag:computer-algebra-systems].

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8
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1answer
61 views

(Un)distorted subgroups in $\mathbb{F}_2 \times \mathbb{F}_2$

We consider the product of free groups $$\mathbb{F}_2 \times \mathbb{F}_2 = \langle a,b,c,d \mid [a,b]=[b,c]=[c,d]=[d,a]=1 \rangle.$$ Given some elements $g_1,\ldots,g_n \in \mathbb{F}_2 \times ...
17
votes
2answers
236 views

Enumerating Bianchi circles

Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi ...
5
votes
2answers
119 views

Algorithm design for enumerating pairs of noncommuting elements up to conjugacy

I am trying to write some Magma code that, given a group $G$, returns a list of pairs $(x,y)$ with $x,y\in G$ such that $[x,y]\neq 1$ and such that every pair $(z,w)$ in the group with $[z,w]\neq 1$ ...
2
votes
0answers
93 views

How to compute/ find cancellation for the second group cohomology $H^2(G,A)$?

My problem is the following, suppose you have a discrete group $G$ (finite type) and a $G$-module $M$, $M$ is a $\mathbb{Q}$-vectorial space. I would like to know if there are "ways" to compute ...
0
votes
1answer
19 views

Ring module homomorphism properties

$\text{Hom}_{\mathbb Z} (\mathbb Z/3\mathbb Z , \mathbb Z/5\mathbb Z) = ?$ how many homomorphism are there from $\mathbb Z/3\mathbb Z$ to $\mathbb Z/5\mathbb Z = ?$ Where $\mathbb Z$ represents ...
2
votes
1answer
47 views

Algorithms for generating $A_n$ and $S_n$ from specific generators

Is there a simple algorithm to generate the elements of the alternating group $A_n$ in terms of some small set of generators? For example, when $n = 4$, I'm looking for an algorithm whose output is a ...
2
votes
1answer
72 views

do we have any special algorithms or software to find all 2-sylow subgroups of a group?

I am working on a project that involves 2-sylow subgroups of groups,one thing that I need to do is to find all 2-sylow subgroups of a group and check that it is cyclic or not, now my question is that ...
6
votes
6answers
292 views

How to efficiently represent and manipulate polynomials in software?

I've started to work on a package (written in matlab for now) that among other things must be able to represent and manipulate (compare, add, multiply, differentiate, etc) polynomials in several ...
0
votes
1answer
73 views

Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
1
vote
1answer
55 views

Write two (or more) numbers as sum of multiples of other numbers (one, two or more)

I have the following problem: Numbers 32, 35 and 57 can be written as sum of multiples of 7 and 9: 32 = (7*2) + (9*2) 35 = (7*5) + (9*0) 57 = (7*3) + (9*4) Is ...
3
votes
0answers
46 views

Software for Braid Groups

I am looking for software/online tool that works on Braid Groups. I am aware that there are resources that allows you to draw braids by imputing generators or detect whether two braids are equivalent ...
0
votes
1answer
33 views

Calculating $b_1,b_2,…,b_k$ where $b_i$=$a_1a_2…a_{i-1}a_{i+1}…a_k$ in minimal number of multiplications

Let's suppose we have a set of integers $a_1, a_2, ..., a_k$ in $Z_n^{*}$, and that we define $b_i$ to be the multiplication $a_1a_2...a_{i-1}a_{i+1}...a_k$. Is there a way to calculate the set ...
0
votes
2answers
95 views

Minimum value of a positive definite binary quadratic form along integers

Is there a formula for the least non-zero value of $$f(x,y):=ax^2+bxy+cy^2$$ as $x,y$ assume integer values? Here $a,b,c$ are integers with $a,d>0$ and $b^2-4ac<0$.
1
vote
0answers
36 views

Quantum Mechanics in Electric Field

I asked this problem in Physics SE but I did not get any useful answers except one. I believe asking this question here would be more beneficial owing to the Mathematical nature of the problem. I am ...
5
votes
2answers
120 views

Conjugacy classes of PSL(6,7)

I need the conjugacy class sizes of projective special linear group PSL(6,7). I couldn't find it by using GAP. Could someone find it?
0
votes
0answers
48 views

Computational cost of solving $Ax_i = b_i$ for $i=1,…,m$

$A$ is an invertible matrix square $n$ matrix. The exercise is about 3 different ways you can solve this and I have to determine its efficiency. It's always the same matrix $A$ but a different right ...
3
votes
1answer
84 views

Computing Images of Varieties

Somehow, this problem has been coming up a lot lately in different guises, which I'm taking as a sign that I ought to stop avoiding computational algebraic geometry. I could probably dig this up in ...
4
votes
1answer
115 views

how to begin self study of computational group theory.

Today in class on finite group theory our professor taught us Mathieu groups and so we dealt with Steiner system and similar. He said from here on you can pursue computational group theory and start ...
4
votes
2answers
92 views

The determination of the Galois group of a polynomial

The GAP package has a function $\mathtt {GaloisType}$ that takes a polynomial as an argument and returns a number, the index of the transitive group of order the degree of the polynomial. I read ...
0
votes
1answer
26 views

Calculating Bloch-Wigner dilogarithm

Is there some tool/calculator (or some tables) for explicitly calculating values of the Bloch-Wigner dilogarithm?
2
votes
1answer
61 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
1
vote
1answer
146 views

Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
9
votes
1answer
177 views

Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
3
votes
1answer
231 views

Is there any efficient algorithm for finding subgroups of a given finite group?

I am implementing an algorithm which finds every subgroup of given group. Here's my algorithm. Let $G$ be a group of order $n$ with elements $g_1,\cdots,g_n$. Then I consider each $\langle ...
1
vote
1answer
69 views

Isomorphism between two magmas with one.

Do we have a method to find one (or all) isomorphism between two given magmas with one using GAP? Edit If we have Loop or Latin square (with one) instead of Magma then do we have the method?
2
votes
1answer
78 views

Let I, J ideals. Are they equal?

Let $$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$ $$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$ ideals. Examine whether those two ideals are equal. By seeing their 3D plots I ...
3
votes
0answers
25 views

Has there been work on computational group theory applications to computing colimits of crosses n-cubes of groups?

I'm trying to compute homotopy groups of a few spaces using crossed n-cubes of groups. I'm able to describe a few colimits in terms of quotients of induced crossed modules and nonabelian tensor ...
0
votes
0answers
64 views

number of symmetries of an arbitrary graph

Given an (undirected) graph G, is there way to (approximately) estimate the order of Aut(G)-- i.e., the number of permutations ...
0
votes
0answers
19 views

Calculus of trigonometric functions based on elliptic Gauss functions?

Considering this 3 concepts: Arithmetic geometric mean Elliptic integral ( in relation to Gauss studies ) Newton's method I'm supposed to be able to write an algorithm to compute trigonometric ...
5
votes
1answer
94 views

Number of subgroups of order $4$ and $8$ in a group of order $72$

Let $G$ be a group of order $72$. I want to calculate the number of subgroups of order $4$ and $8$ with GAP. How can I do? thanks in advance.
4
votes
1answer
182 views

How to find presentation of a group using GAP?

I have a group from Small Group Library and I want to find its presentation using GAP. I have tried to use PresentationFpGroup(G) but failed. Please suggest me a method.
0
votes
0answers
55 views

Boosting of Coset diagrams

If we have the diagram that represents a transitive permutation representation of $(p,q,r_o)$ for some $p, q$ and $r_o$, we often use this diagram to get diagrams for any $r>r_o$. We can do this ...
2
votes
2answers
151 views

Does the Windows version of MAGMA have a memory limit?

Sorry if this isn't the best place to ask support type MAGMA questions, but I haven't found a single forum or anything for MAGMA users to talk. I have access to a copy of MAGMA which is running on a ...
2
votes
1answer
85 views

Expressing a matrix in terms of subgroup generators using Magma

With Magma, it is possible to define a subgroup $H$ of a finite matrix group $G$ in terms of generators. Given a matrix $M\in G$, Magma can also determine whether $M\in H$. Presumably, if Magma ...
2
votes
0answers
46 views

An algorithm for generating a finite group with a finite set of generators

Let $A$ be a finite set of permutations on $\Bbb N$ with finite support. Is there a good efficient algorithm to obtain the subgroup $\left<A\right>$ of the symmetric group ...
4
votes
3answers
145 views

Galois group of $x^5-12x+2$ over $\mathbb{Q}$

I've always been able to compute the Galois groups of polynomials of degree $\leq 4$, but I have trouble at higher degrees. I can factor quadratics and cubics, and get the solutions from there, but ...
2
votes
1answer
44 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
4
votes
2answers
254 views

Find the number of irreducible factors of $x^{63} - 1$

I have to find the number of irreducible factors of $x^{63} - 1$ over $\mathbb{F}_2$ using the $2$-cyclotomic cosets modulo $63$. Is there a way to see how many the cyclotomic cosets are and what is ...
1
vote
0answers
76 views

Does this proof that the resultant provides an upper bound for intersection multiplicity look correct?

Let $f,g \in \mathbb{C}[x,y]$ such that $f(0,0)=g(0,0)=0$ and the varieties $V(f)$ and $V(g)$ are both smooth at $(0,0)$ such that the tangent line of $V(f)$ and $V(g)$ is not the $y$ axis. Let ...
2
votes
0answers
58 views

The Command “TzGoGo” in GAP

I am learning GAP and would like to ask one question about a command called "TzGoGo": If $P$ is a finite presentation of a group $G$, then will the eventual result of the command "TzGoGo(P)" be ...
3
votes
1answer
101 views

GAP Responding Time

I am running GAP 4.6.5 on my six-year-old computer and sometimes it takes like forever for GAP to respond to my simple commands. An easy example is as follows: ...
3
votes
0answers
132 views

Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked ...
2
votes
0answers
135 views

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated ...
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votes
0answers
52 views

minmax of hamming weight in a basis for a vector space

Consider the vector space $V=\mathbb{Z}_2^n$ and take some linear subspace $U\subseteq V$ (we can assume we are given some basis for $U$). Now, for every basis $B$ of U, define $f(B)=\max_{b\in ...
3
votes
1answer
124 views

Algorithm for Finitely Presented Torsion-Free Nilpotent Groups

I am studying some finitely presented, torsion-free and nilpotent groups $G$ and need to consider the following question: Let $H$ be a subgroup of $G$ and suppose that $H$ is generated by ...
2
votes
3answers
109 views

Websites/Software for Group computation

Anyone knows a website or software that helps to do computations in a group? For example, by inputting generators and relations in the group, can we tell when two particular elements in the group ...
0
votes
1answer
97 views

Efficient way to compute min/max

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-\tfrac12 a_1 + a_1 b_1+\tfrac12 a_1 b_2-\tfrac12 a_1 a_2 b_2 + 2 a_1 a_2 - \tfrac32 a_1 a_2 b_1 - \tfrac12 a_4 + a_4 b_1 + ...
0
votes
1answer
82 views

Do we need Gröbner bases to study factor rings of polynomials?

I'm trying to understand how we can systematically study the factor rings of polynomials over a ring K. For example imagine that we're working in $R=K[x_1,...,x_n]$ and we have the ideal ...
1
vote
1answer
83 views

Product of (strongly) stable ideals and lexsegment ideals

(1) Is the product of lexsegment ideals again a lexsegment ideal? (2) Is the product of (strongly) stable ideals again (strongly) stable? I know that both of them are false and I can find examples ...
16
votes
1answer
472 views

What is computational group theory?

What is computational group theory? What is the difference between computational group theory and group theory? Is it an active area of the mathematical research currently? What are some of the ...