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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22
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1answer
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 ...
6
votes
1answer
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: ...
16
votes
1answer
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 ...
7
votes
1answer
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 ...
3
votes
0answers
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 ...
4
votes
2answers
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 ...
3
votes
2answers
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 ...
3
votes
3answers
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 ...
2
votes
0answers
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 ...