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 ...

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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 ...
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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 ...
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56 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 ...
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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 ...
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66 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.
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71 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.
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40 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 ...
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60 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 ...
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50 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 ...
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34 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 ...
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106 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 ...
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39 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 } ...
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208 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 ...
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40 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 ...
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43 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 ...
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76 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: ...
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49 views

Urgently Seeking for Help in Calculating the Abelian Invariants of a Group with GAP

I have asked a more complicated question here but I decided to start with an easier version: Let $K_3$ be the group generated by three elements $a$, $b$, $c$ subject to the relation that every simple ...
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112 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 ...
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94 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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40 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 ...
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90 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 ...
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3answers
97 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 ...
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91 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 + ...
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71 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 ...
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78 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 ...
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366 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 ...
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95 views

The h-vector of a simplicial complex

Let $S$ be a polynomial ring over a field. I want to find an ideal $ I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$. We have a relation between $f$-vector and $h$-vector and ...
4
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1answer
79 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 ...
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154 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 ...
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102 views

Listing subgroups of a group

I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as ...
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231 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 ...
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151 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 ...
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136 views

Necessary and sufficient conditions for Hensel lifting in the multidimensional case

in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
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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 ...
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189 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
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1answer
264 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 ...
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138 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 - ...
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2answers
97 views

Mutiple root of a polynomial modulo $p$

In my lecture notes of algebraic number theory they are dealing with the polynomial $$f=X^3+X+1, $$ and they say that If f has multiple factors modulo a prime $p > 3$, then $f$ and $f' = ...
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638 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 ...
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1answer
446 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 ...
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1answer
221 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 ...
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311 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: ...
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71 views

First order logic in polynomial equations

Have you ever wondered which points on a conic are the intersections of tangent lines of another surface through the origin? More generally, which points on a shape hold some specified relation to all ...
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82 views

How to tell if an ideal is absolutely prime

Consider the ideal $I=(ag-ec-1,ah+bg-cf-de)$ of $R=K[a,b,c,d,e,f,g,h]$. Is $I$ prime when $K=\overline{\mathbb{F}}_2$ is the algebraic closure of a field of 2 elements? Can computers answer this ...
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164 views

How to show some systems of equations do not have a closed form solution?

How to show some systems of equations do not have a closed form solution? for example I was once given something similar to ( this might not be the exact problem but i am just using it as an example ...
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224 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 ...
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252 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 ...