A section of a group G is a quotient of a subgroup, viz. K normal in H subgroup of G, the section is H/K.
Does anyone know if there is a more-or-less standard terminology not just for the section-up-to-isomorphism-of-the-group-H/K, but for the "instantiated" or maybe "concrete" section, e.g. the pair (K,H)?
algebra final tmrw 😤 watch me cook
Quanta Magazine really has nicely in-depth yet approachable writing on #mathematics, in this case an introduction to Lie groups #GroupTheory https://www.quantamagazine.org/what-are-lie-groups-20251203/
Some time ago for practicing #haskell I implemented a well know algorithm for combining multiplets https://github.com/mdrslmr/MultipletCombiner . This algorithm from #grouptheory is used in #particlephysics and other #physics topics. The code is probably pretty useless since the results are anyway textbook standards. But I found it was a nice exercise. I'm sure the code can be improved a lot easily.
Một lập trình viên đã tối ưu tốc độ thư viện đường cong Elliptic, chỉ sử dụng lý thuyết nhóm thuần túy trong toán học, mà không cần đến lập trình cấp thấp (assembly) hay tăng tốc GPU (CUDA). Đây là một thành tựu đáng chú ý trong việc cải thiện hiệu suất mật mã.
#EllipticCurve #Cryptography #GroupTheory #Optimization #Math #Programming #ĐườngCongElliptic #MậtMãHọc #LýThuyếtNhóm #TốiƯu #ToánHọc #LậpTrình
https://www.reddit.com/r/programming/comments/1nyrnmt/if_youre_so_smart_then_why_are_you_poor
Evariste #Galois - #GroupTheory #QuantaMagazine #Maths #Mathematics
#GaloisGroups and the #Symmetries of #Polynomials
By Allison Whitten
Why are normal subgroups called "normal?" Most subgroups are not normal, which seemingly makes the name incorrect.
But in an Abelian group, all subgroups are normal. Abelian groups were the first groups to be studied by white European men, and they are the easiest for humans to work with.
Therefore, normal subgroups are called normal because they are the subgroups which pose the least challenge to the incumbent power structures.
#GroupTheory #NormalSubgroup
Petra Cini, composer and pianist, gave a special #nikhef theory seminar on musical representations: violence, purity and mathematics
Drafted a 3rd inquiry activity for D3 and D4 play with flips and rotations.
I am hoping it's not to dry. I'll read through it again tomorrow.
#DocMadhattan: The Abel Prize 2025: Masaki Kashiwara
https://docmadhattan.hashnode.dev/the-abel-prize-2025-masaki-kashiwara
Masaki Kashiwara wins the 2025 Abel Prize for groundbreaking work in algebraic analysis, D-modules theory, and crystal bases in representation theory
#mathematics #physics #AbelPrize #MasakiKashiwara #grouptheory #quantumgroups
I've kind of always wondered what the point of definitions like a group is a non-empty set \(G\) with a binary operation \(d\) satisfying \(d(d(d(z,d(x, d(x,x))),d(z,d(y,d(x,x)))),x) = y\) is, other than because we can, but https://math.stackexchange.com/a/4366021 offers one such answer in terms of homotopy type
searching for any structures / theory that involve a particular operation on non-empty lists of postitive integers like "the length of the list multiplied by the least common multiple of all the items in the list"
any ideas? references to any literature would be very appreciated if you know of any.
#askfedi #math #maths #mathematics #combinatorics #GroupTheory #DiscreteMath
Me: If you're a student in my class you will do group presentations, meaning that you will organize into a subset of the class, stand in front of the room, and talk about math to the rest of the class.
Also me: Let's talk about group presentations, a completely abstract algebraic concept that have absolutely nothing to do with standing in front of the room talking about your work.
Also also me: Okay, time for group presentations about group presentations.
#Algebra #ITeachMath #GroupTheory #GroupPresentation
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existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way.
einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral.[3] Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically.
Partition of a plane in closed set - tile
2022, hobbyist David Smith discovered a "hat"-shaped tile formed from eight copies of a 60°–90°–120°–90° kite (deltoidal trihexagonals), glued edge-to-edge, which seemed to only tile the plane aperiodically.[8] Smith recruited help from mathematicians Craig S. Kaplan, Joseph Samuel Myers, and Chaim Goodman-Strauss, and in March 2023 the group posted a preprint proving that the hat, when considered with its mirror image, forms an aperiodic prototile set.[
Wiki
#grouptheory
What is a good website to teach Group Theory and Chemistry? #Chemistry #Mathematics #Spectroscopy #GroupTheory
New 📚 Release! Introduction to Group Theory: An Activity-Based Approach by Joe Fox #books #ebooks #math #grouptheory
This book is an introduction to group theory suitable for an introductory course in abstract algebra. Much of the content is delegated to a series of activities that are meant to be worked through by the students with the help of the instructor.
Find it on Leanpub!
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)
\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)
No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png
Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png
Reference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes
#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes
Want to formalize something in Mizar but don't know where to begin?
I'm starting a new series of posts with project ideas, starting with Loops! They're needed to formalize the sporadic groups (as found in, e.g., Aschbacher's book "Sporadic Groups").
I have been formalizing finite group theory in Mizar, and I am about to formalize representation theory.
So I thought I would write some "notes to myself" about it.
#proofassistant #grouptheory #RepresentationTheory #Mizar
https://thmprover.wordpress.com/2024/12/13/plans-for-representation-theory-in-mizar/
A shop called "Wreath Products" that sells mathematical puzzle toys, mobiles (like for baby cribs or art)...and, sure, also decorative wreaths.
