Articles of computational algebra

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 generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

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 here, I am trying to use GAP to do some calculation with $K$. Now one problem is that there are too many relations. For example, all following […]

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.

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 their cardinality which is faster than the direct computation? Thank you.

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 background to study it?

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 the group of Möbius transformations with coefficients in the ring of integers of $K$. The image of $\mathbb R$ under a group element is called a […]

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: gap> C3:=CyclicGroup(IsPermGroup,3); Group([ (1,2,3) ]) gap> C7:=CyclicGroup(IsPermGroup,7); Group([ (1,2,3,4,5,6,7) ]) gap> A:=AutomorphismGroup(C7); < group with 1 generators > gap> elts := Elements(A); [ IdentityMapping( Group([ (1,2,3,4,5,6,7) […]

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 example would help greatly!

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 other words, what is the best way to use […]

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 gap>Elements( Group( (1,2,3,4,5,6,7,8),(1,2) ) ); so fast list all $8!$ elements of $S_8$? Can someone describe the algorithm or the method used?