# Tagged Questions

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].

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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 ...
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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 ...
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### 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 ...
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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 ...
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### 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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### GAP Most efficient way to check multiple properties of a group in the small group library

In GAP I would like to search the small groups library looking for groups with specific properties (I suppose this is the most common usage). If I have a list of properties I want to test, what is ...
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### 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 ...
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 ...
Let $G$ be a finite group with a fixed subset $A$. Put $$S_r(A)=\{B\subseteq G : |AB|=|A||B|\; \& \; B \; \mbox{is inclusion-maximal with respect to this property}\}$$  M_r(A)=\max\{|B| : B\...