Articles of computational algebra

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 example, when $G=\mathbb{Z}_2^n$ and $H=\mathbb{Z}_2$, a homomorphism $f$ is entirely characterized by an element of $\mathbb{Z}_2^n$, $s\in \{0,1\}^n$, such […]

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 applied on this polynomial. I know that one can use factor command in Wolfram Alpha and show that polinomial is irreducible but that isn’t point […]

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 have been a software engineer for past many years. Ideally, I would like an algebra book […]

About subsets of finite groups with $A^{-1}A=G$ or $AA^{-1}=G$?

Regarding to the problems Does $A^{-1}A=G$ imply that $AA^{-1}=G$? and Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?, we are looking for some subsets $A$ of $G$ with $\lfloor \frac{|G|}{2}\rfloor-2 \leq |A|\leq \lfloor \frac{|G|}{2}\rfloor$ such that $A^{-1}A=G$ and $AA^{-1}\neq G$ or $AA^{-1}=G$ and $A^{-1}A\neq G$. Do such $G$ (non-abelian) and $A$ exist? We propose the […]

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 […]