Questions tagged [abstract-algebra]
Abstract algebra is the study of algebraic structures, including groups, rings, fields, vector spaces, and the like.
76 questions
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The monoids with three elements
Objective
There are seven monoids with three elements, up to isomorphism. Give implementations to all of them, such that their domains are all the same, and that they have the same identity element.
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4
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Find a primitive polynomial
Objective
Given a prime number \$p\$ and an integer \$n \geq 2\$, find a degree-\$n\$ primitive polynomial modulo \$p\$.
Mathematical explanation
When we perform "modular arithmetic" over ...
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votes
9
answers
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Output the symmetric inverse semigroup
The symmetric inverse semigroup is a very important object in the study of semigroups, for a number of reasons, but most obviously due to the Wagner-Preston theorem. In brief, for any set \$X\$, the ...
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Free Kei Friday
A kei (圭) is an algebraic structure that abstracts the idea of mirror reflections.
The kei is given as a set of mirrors \$X\$ and a closed reflection operation \$(\rhd) : X\times X\rightarrow X\$. We ...
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votes
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(-a) × (-a) = a × a
We all know that \$(-a) \times (-a) = a \times a\$ (hopefully), but can you prove it?
Your task is to prove this fact using the ring axioms. What are the ring axioms? The ring axioms are a list of ...
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votes
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How does the square end?
In base-10, all perfect squares end in \$0\$, \$1\$, \$4\$, \$5\$, \$6\$, or \$9\$.
In base-16, all perfect squares end in \$0\$, \$1\$, \$4\$, or \$9\$.
Nilknarf describes why this is and how to work ...
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9
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Primitive words
Given a list of values, 1, 2, -1, or -2, we will allow the following simple moves:
Remove adjacent values which are negatives of each other. e.g. ...
- 104k
12
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18
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Normal Subgroups of \$S_4\$
Objective
Given a permutation of 4 distinct items, classify the permutation by the normal subgroup(s) it belongs.
Input/Output Format
You gotta choose the followings as the hyperparameters for your ...
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14
votes
5
answers
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Generate a subgroup of a free group
In group theory, the free group with \$n\$ generators can be obtained by taking \$n\$ distinct symbols (let's call them \$a, b, c ...\$ etc), along with their inverses \$ a^{-1},b^{-1},c^{-1} ...\$ . ...
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8
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4
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Determine if two trees are equal in the free quandle
To start we are going to define an "\$\operatorname{FBM}\$" as follows:
Every integer is an \$\operatorname{FBM}\$.
If \$a\$ and \$b\$ are \$\operatorname{FBM}\$s, then \$a \lhd b\$ is an \$...
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17
votes
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Construct this point
Given a constructible point \$(x, y) \in \mathbb R^2\$, output the steps required to construct \$(x, y)\$
Constructing a point
Consider the following "construction" of a point \$(\alpha, \...
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24
votes
25
answers
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Multiply elements of the dihedral group
This is a copy cat question of Simplify ijk string applied to the other nonabelian group of order 8. See also Dihedral group composition with custom labels.
Challenge
Given a string made of ...
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7
votes
1
answer
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Solving the high school algebra problem
We learned many identities involving addition, multiplication and exponentiation in highschool, for example:
$$ \begin{aligned}
(a+b)c &= ac + bc \\
(a b)^c &= a^c b^c \\
(a^b)^c &= a^{bc}
...
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27
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27
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Modular multiplicative inverse
Your task is to given two integer numbers, a and b calculate the modular multiplicative inverse of a modulo b, if it exists.
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Count the symmetries
Find the order (size) of the symmetry group of a finite set of integer points in d-dimensional space.
Input
You will be given the coordinates of a finite set of points in d-dimensional space, in any ...
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