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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9
votes
4
answers
757
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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 ...
- 7,583
9
votes
2
answers
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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
votes
9
answers
889
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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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6
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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 ...
- 104k
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}
...
- 805
8
votes
4
answers
432
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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 \$...
- 104k
6
votes
2
answers
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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 ...
- 2,082
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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17
votes
2
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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, \...
- 51.1k
8
votes
0
answers
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Solve the word problem*
In this challenge you will receive a list of positive integers \$W\$ called a word, and a square symmetric matrix \$M\$. Your task is to determine if the word can be turned into the empty list by ...
- 104k
14
votes
1
answer
355
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Construct the Constructability sequence
Consider compass-and-straightedge construction, where you can construct new points from existing ones by examining intersections of straight lines and circles constructed with one of the following two ...
- 51.1k
16
votes
6
answers
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Detect round trips on a hyperbolic grid
You're driving a car in an infinite city whose blocks are pentagons arranged in the order-4 pentagonal tiling. At each step, you proceed to the next intersection and choose whether to continue left, ...
- 871
12
votes
18
answers
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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 ...
- 7,583
10
votes
20
answers
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Give the inverse permutation
Task
Given a finite permutation output its inverse.
You may take input and output in any reasonable format equivalent to a list of natural numbers. You may choose to use 0 indexing or 1 indexing. ...
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