# 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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### Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
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### Groebner basis over rings

Let $I$ be an ideal in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian commutative ring, such that w.r.t some monomial order it has a Groebner basis $G = \{g_1, \ldots, g_t\}$ with all the leading ...
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### An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
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### Algebraic description of $\frac1{x+y}$ in terms of $\frac1x$ and $\frac1y$ over finite field.

Given an $a = \frac1x$ and $b = \frac1y$, is there some algebraic way to get the value $c = \frac1{x+y}$ using $a$ and $b$ without the use of inversions. I can't seem to figure it out and it might be ...
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### How to find a base of a permutation group?

How to find a base of the permutation group G=âŸ¨x,yâŸ©â‰¤S4 x=(1,2,3), y=(1,2,4)? I hear that base for G is a sequence B = [$b_1, ..., b_m$] âŠ‚ â„¦ such that the only element of G which stabilizes each $b_i$ ...
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### 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 ...
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### Order of a group given its generating set

Let $n,k \in \mathbb{N}$ and $x_1, \dots, x_k \in S_n$, symmetric group. Is there an efficient algorithm to determine the $$\text{order of }\langle x_1, \dots, x_k\rangle\text{?}$$ If necessary, ...
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### Todd-Coxeter algorithm: coincidences

I'm trying to understand the Todd-Coxeter algorithm with the help of a multiplication and relator table, but there is one thing about coincidences that is not really clear. For some small groups (for ...
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### Is there always a non-cyclic abelian subgroup of S2n with order n^2?

Let $n \in N$. Does there always exist a non-cyclic abelian subgroup of $S_{2n}$, the symmetric group on 2n letters, with $n^2$ elements? How could a computer give an example of such subgroup in? ...
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I am studying the Mathieu group $M_{11}$ on the twelve letters $\infty,7,6,8,X,2,0,3,4,1,9,5$ (in this specific order) in the form that it is generated by the permutations $(0123456789X)$, $(13954)(... 1answer 77 views ### Given the generators, find the entire group my question is quite simple. If you are given the generators of a group, is there any systematic way to generate all of the elements of the group? For example, suppose that you have the Hadamard ... 0answers 35 views ### Write an element of a group as product of generators Say we have a finite group$G$generated by$g_1,\cdots,g_n$. Are there any algorithms or techniques to write any element$g$as a product of these generators? Ofcourse we could just try all ... 0answers 31 views ### Algorithmic computing kernel of a graded homomorpism For computing kernel of a module homomorphism we can use module-Grobner basis such as described in notes talking about computing SyZyGies. How can we compute kernel of a homomorphism between a graded ... 1answer 26 views ### Expected number of rows of the full rank matrix Let A be a m by n random matrix over finite fields F_q. Suppose the rank of A is n. How much does expected number of m? I think m maybe qlogq by bins and balls property But I do not know exactly why.... 3answers 157 views ### How do Gap generate the elements in permutation groups? I understand that permutationgroups in Gap are represented by generators, which seems to be far more efficient than groups represented by all it's elements, but how could then for example ... 2answers 308 views ### Number of Irreducible Factors of$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 ... 1answer 48 views ### Isomorphic subgroups of finite groups Which is the smallest number$n$such that$S_n$has non-isomorphic subgroups of the same order with the same number of cyclic subgroups of the same order? Example:$S_4$has subgroups of order$...
As I understand There can't be a general algorithm to decide if two finite groups are isomorphic, Wikipedia. But are there efficient algorithms for all subgroups of $S_n$ for say $n=10$ or so? ...