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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23
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1answer
784 views

Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How ...
18
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1answer
722 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 ...
17
votes
2answers
286 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 ...
15
votes
1answer
379 views

Most wanted reproducible results in computational algebra

I am interested in suggestions for major computational results obtained with the help of mathematical software but not easily verifiable using computers. "Most wanted" could refer, for example, to ...
14
votes
1answer
161 views

The equivalence classes of $N\sim M\Leftrightarrow G/N\cong G/M$.

Let $G$ be a finite group. Given some $N\unlhd G$, define $$\mathfrak{C}_N:=\{M\unlhd G : G/M \cong G/N\}.$$ How are the subgroups in $\mathfrak{C}_N$ related? Is there some other description ...
9
votes
1answer
207 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 ...
8
votes
2answers
296 views

Is it “often known” how to compute a list of groups?

From the introduction of the article Construction of Finite Groups written by Hans Ulrich Besche and Bettina Eick: When attempting to determine up to isomorphism the groups of a given order it is ...
8
votes
1answer
265 views

Applications of computation on very large groups

I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all ...
8
votes
1answer
92 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 ...
7
votes
3answers
136 views

How do Gap generate the elements in permutation groups?

I understand that permutationgroups in Gap are represented by generators, which seems to be far more efficient than groups represented by all it's elements, but how could then for example ...
7
votes
6answers
800 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 ...
7
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1answer
873 views

Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?

I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
6
votes
2answers
303 views

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 ...
6
votes
2answers
491 views

Lower central series of a free group

Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding ...
6
votes
1answer
456 views

GAP semidirect product algorithm

Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group? EXAMPLE: ...
5
votes
1answer
501 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.
5
votes
2answers
229 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 $[z,w]\neq 1$ ...
5
votes
1answer
106 views

Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example?

Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example? All the books I read have a dense notation that hard to comprehend but a simple and concrete ...
5
votes
2answers
132 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?
5
votes
1answer
177 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 ...
5
votes
1answer
117 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.
5
votes
2answers
368 views

Check whether two subgroup of $GL(n,\mathbb Z)$ are conjugate

Suppose I have two finite subgroups of $GL(n,\mathbb Z)$. Is there an algorithm to find out whether these two belong to the same conjugacy class in $GL(n,\mathbb Z)$? I tried by using the Jordan ...
5
votes
1answer
655 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 ...
4
votes
2answers
246 views

How to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers?

Which criterion (test) one can use in order to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers ? Neither of Eisenstein's criterion and Cohn's criterion cannot be ...
4
votes
2answers
112 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 ...
4
votes
1answer
86 views

Generators of a subgroup of $SL_2(\mathbb{Z}/24\mathbb{Z})$

So I have this subgroup of $SL_2(\mathbb{Z}/24\mathbb{Z})$ which has $256$ elements. Is there a way in sage to get the list of its generators ? The "only" information I have on the group is the list ...
4
votes
3answers
170 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 ...
4
votes
3answers
138 views

Does $1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$ have a global minimum?

Does the following function have a global minimum: $$1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$$ where $x \in \mathbb{N}$? I tried using WolframAlpha, but it appears to give an inconsistent ...
4
votes
4answers
141 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
4
votes
1answer
68 views

GAP Most efficient way to check multiple properties of a group in the small group library

In GAP I would like to search the small groups library looking for groups with specific properties (I suppose this is the most common usage). If I have a list of properties I want to test, what is ...
3
votes
3answers
175 views

Software for deciding ideal membership

Let $\alpha$ be such that $\alpha^3 + \alpha + 1 = 0$ and consider $\Bbb{Z}[\alpha]$. Suppose I have an ideal in $\Bbb{Z}[\alpha]$ that is given by $$ I = \Bigg(23^3, 23^2(\alpha - 3), 23(\alpha - ...
3
votes
2answers
414 views

Converting GAP groups into SAGE permutation groups.

I have been working with SAGE online, and have made some programs to test some hypothesis about finite groups. However, the pre-defined "named" groups in SAGE are quite limited (basically, the ...
3
votes
2answers
103 views

Is there an easy way to find the (number of) subgroups of a given group?

Is there an easy way to find all subgroups of a given group? For example, say you had the dihedral group $D_{4}$ - is there a way to work out how many subgroups this has, or what they are, or can it ...
3
votes
1answer
97 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 ...
3
votes
1answer
70 views

Given the generators, find the entire group

my question is quite simple. If you are given the generators of a group, is there any systematic way to generate all of the elements of the group? For example, suppose that you have the Hadamard ...
3
votes
1answer
92 views

Do circular tapes exist in Turing Machines?

I've been looking for information about this topic without success. Have someone described Turing Machines over circular tapes instead lineal and infinite? Like the tape could be described with ...
3
votes
1answer
32 views

Algorithmic way to check if a power-conjugate presentation is consistent?

Is there an algorithmic way to check if a power-conjugate presentation (for a finite polycyclic group) is consistent? Background: A finite solvable group $G$ has a subnormal series $$ G=G_0 ...
3
votes
1answer
92 views

Equation over free group

Let us consider free group $F(a,b)$ of rank two. I need to find a solution (or to prove that there is no one) over this group of the following equation: $$x^2[x^{2k},y]=a^2b^2,$$ where ...
3
votes
1answer
95 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 ...
3
votes
1answer
120 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
1answer
185 views

Permutation calculator

I am studying the Mathieu group $M_{11}$ on the twelve letters $\infty,7,6,8,X,2,0,3,4,1,9,5$ (in this specific order) in the form that it is generated by the permutations $(0123456789X)$, ...
3
votes
1answer
143 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 ...
3
votes
1answer
403 views

Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$.

Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables. Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the ...
3
votes
3answers
276 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
3
votes
0answers
43 views

Software for computing generators of the invariant rings of the symmetric groups

(Please skip to the last paragraph if you are interested in just the question) I wish to compute the generators of the ring of invariants for a symmetric group acting on a polynomial ring using a ...
3
votes
0answers
23 views

Algorithmic computing kernel of a graded homomorpism

For computing kernel of a module homomorphism we can use module-Grobner basis such as described in notes talking about computing SyZyGies. How can we compute kernel of a homomorphism between a graded ...
3
votes
0answers
82 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 ...
3
votes
0answers
26 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 ...
3
votes
0answers
155 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
3answers
127 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 ...