Questions tagged [computational-algebra]

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].

Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields, etc.), such as, for example, deciding whether two groups are isomorphic, finding the decomposition of a group element in terms of the group generators, computing the lattice of subgroups of a group, etc. (see, for example, http://maths.nuigalway.ie/deBrun6/Brooksbank/Galway-1.pdf).

Thus, computational algebra differs from computer algebra, since the latter deals with questions like carrying out symbolic manipulations with mathematical expressions (though, of course, practical implementations may overlap in some areas).

Please use these notes to distinguish between the tag for questions about generic computer algebra systems, and the tag for questions about the topics outlined above.

267 questions
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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 difficult is it to recover the group structure of $G$? In…
24
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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 the following: results which are highly…
22
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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 most interesting results? What is the needed…
17
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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 group $\mathrm{PSL}_2(\mathcal{O}_K)$, which is…
14
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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 of $\mathfrak{C}_N$? Would there be a more direct way…
11
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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 which cover Galois theory (and its applications)? I…
11
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Sum of determinants of block submatrices

I have a $2n \times 2n$ matrix, $M$. I view it a block matrix, of $n^2$ blocks, each of shape $2\times 2$. Computing the determinant of $M$ is easy by conventional methods. I could also look at diagonal block submatrices: given a set $S \subseteq…
11
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Is this specific group finite?

I have the following group presentation: $G=\left\langle a,b,c\ |\ a^2,b^{11},c^2,(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},(ac)^3,(bc)^2\right\rangle$ Is $G$ finite? GAP's Size(G) runs out of memory pretty quickly. No surprise there. I…
Josh B.
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9
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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.
Shodharthi
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9
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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 properties of extremely large groups, moreover it…
9
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Is there a group with $2$ generators having exactly $17$ subgroups of index three?

I recently saw a fun problem from a past qualifying exam from Stanford. It is Problem 10, part (b) in this document. I will screenshot the problem and its solution here: My question is the following. Does there exist a group $G$ which is…
9
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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 variables. Up until now my approach has been the…
jkn
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8
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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 often known how to compute a list of groups of…
8
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Why is it so computationally hard to determine group isomorphism?

Finding an isomorphism requires to show that for 2 groups $G$ and $H$, there exists a bijective map $\phi : G\to H$ such that $$\phi(ab)=\phi(a)\phi(b)$$ For all $a,b \in G$. This is (probably naively) pretty straight forward, and there are plenty…
8
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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 $G\le \text{P}{\Gamma}\text{L}(n,q)$, find its…
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