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19
votes
1answer
434 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 ...
13
votes
1answer
109 views
The equivalence classes of $N\sim M\Leftrightarrow G/N\cong G/M$.
Let $G$ be a finite group. Given some $N\unlhd G$, define $$\mathfrak{C}_N:=\{M\unlhd G : G/M \cong G/N\}.$$
How are the subgroups in $\mathfrak{C}_N$ related? Is there some other description ...
5
votes
1answer
190 views
Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?
I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
5
votes
0answers
141 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:
...
4
votes
2answers
185 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 ...
4
votes
2answers
86 views
Does $1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$ have a global minimum?
Does the following function have a global minimum:
$$1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$$
where $x \in \mathbb{N}$?
I tried using WolframAlpha, but it appears to give an inconsistent ...
4
votes
1answer
46 views
Lower central series of a free group
Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding ...
3
votes
1answer
183 views
Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$.
Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables.
Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the ...
3
votes
3answers
220 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
3answers
102 views
Software for deciding ideal membership
Let $\alpha$ be such that $\alpha^3 + \alpha + 1 = 0$ and consider $\Bbb{Z}[\alpha]$. Suppose I have an ideal in $\Bbb{Z}[\alpha]$ that is given by
$$ I = \Bigg(23^3, 23^2(\alpha - 3), 23(\alpha - ...
2
votes
2answers
49 views
Mutiple root of a polynomial modulo $p$
In my lecture notes of algebraic number theory they are dealing with the polynomial $$f=X^3+X+1, $$ and they say that
If f has multiple factors modulo a prime $p > 3$, then $f$ and $f' = ...
1
vote
1answer
128 views
Converting GAP groups into SAGE permutation groups.
I have been working with SAGE online, and have made some programs to test some hypothesis about finite groups. However, the pre-defined "named" groups in SAGE are quite limited (basically, the ...
1
vote
0answers
68 views
Necessary and sufficient conditions for Hensel lifting in the multidimensional case
in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
1
vote
0answers
63 views
First order logic in polynomial equations
Have you ever wondered which points on a conic are the intersections of tangent lines of another surface through the origin? More generally, which points on a shape hold some specified relation to all ...
1
vote
0answers
73 views
How to tell if an ideal is absolutely prime
Consider the ideal $I=(ag-ec-1,ah+bg-cf-de)$ of $R=K[a,b,c,d,e,f,g,h]$.
Is $I$ prime when $K=\overline{\mathbb{F}}_2$ is the algebraic closure of a field of 2 elements?
Can computers answer this ...
0
votes
1answer
119 views
How to show some systems of equations do not have a closed form solution?
How to show some systems of equations do not have a closed form solution?
for example I was once given something similar to ( this might not be the exact problem but i am just using it as an example ...
0
votes
0answers
75 views
An Algorithm to Find the Generators of the Radical of a Monomial Ideal
Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
0
votes
0answers
202 views
Simple Lanczos algorithm code to obtain eigenvalues and eigenvectors of a symmetric matrix
I would like to write a simple program (in C) using Lanczos algorithm. I came across a Matlab example which helped me to understand a bit further the algorithm, however from this piece of code I can't ...
