What are good references for #GroupTheory beyond the introduction? @ProfKinyon?

#Math

More older notes rehabilitation, this time on group theory.

My notes are here: https://davidmeyer.github.io/qc/groups.pdf. The LaTeX source is here: https://www.overleaf.com/read/xjmnmzvrgsfk. As always, questions/comments/corrections/* are greatly appreciated.

#math #maths #grouptheory #firstisomorphismtheorem #TeXLaTeX

One of my favorite theorems in all of group theory is the beautiful Cayley's Theorem, which is named in honor of British mathematician and lawyer Arthur Cayley. Cayley was born #onthisday 202 years ago.

Cayley was the first person to define the concept of a group in the modern (and general) way, that is, as a set with a binary operation satisfying certain laws [1]. Before Cayley when mathematicians talked about "groups" they meant permutation groups.

Cayley had many other mathematical accomplishments. For example, Cayley postulated the Cayley–Hamilton Theorem, which states that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3 [2]. Cayley graphs and Cayley tables are also named in his honor.

A few of my notes on Cayley and Cayley's Theorem are here: https://davidmeyer.github.io/qc/cayleys_theorem.pdf. The LaTeX source is here: https://www.overleaf.com/read/nbfyqkwsfmyc.

As always, questions/comments/corrections/* greatly appreciated.

#math #maths #grouptheory #cayleystheorem

References
--------------
[1] "Arthur Cayley", https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/

[2] "Cayley–Hamilton theorem", https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem

@j2kun So addition and negation, when operating on the 𝑛-bit integers, generate the dihedral group (https://en.wikipedia.org/wiki/Dihedral_group) that expresses the symmetries of a \(2^{n-1}\)-gon.

I didn't think of this before. 😃

#GroupTheory

I have moved this little project to:

https://github.com/mdrslmr/MultipletCombiner

https://hackage.haskell.org/package/MultipletCombiner

https://aur.archlinux.org/packages/haskell-multipletcombiner

#haskell
#physics
#grouptheory
#math
#functionalprogramming

My last math class in college: Modern Algebra. #111Words #Math #ModernAlgebra #GroupTheory #GregoryBrumfiel #Stanford https://andrewjshields.blogspot.com/2023/05/my-last-math-class-in-college-modern.html

Ah!
Jamie Mulholland of Simon Fraser University, the one with the book "Permutation Puzzles, A Mathematical Perspective", has the lecture about that topic online!

https://www.youtube.com/playlist?list=PLKXCdnugmHRm2ICudfhC9xco1p350mma2

#GroupTheory #groups #math #puzzles

From Wikipedia:
"Any group G is the homomorphic image of some free group F"
https://en.wikipedia.org/wiki/Free_group#Facts_and_theorems
AFAIK the reverse is true as well (provided that you only look at )

Something like:
"If you have a set S and a group F, such that any other group that has S as its set of generators is a homomorphic image of F, then F is the free group for the set S"

#categorytheory #grouptheory

I've been trying to draw this in an easy to look at way. This is what I came up with.

My notes are here: https://davidmeyer.github.io/qc/linear_algebra.pdf

As always, questions/comments/corrections/* greatly appreciated.

#linearalgebra #grouptheory

I've been working a bit on some notes about the relationship between linear maps and group homomorphism, as well as how matrices induce linear maps.

My (nascent) notes are here: https://davidmeyer.github.io/qc/linear_algebra.pdf.

As always, questions/comments/corrections/* greatly appreciated.

#math #linearalgebra #grouptheory

A weird thing: For considering \( (\mathbb Q, +) \) as a group, one must consider the knowledge of the whole operations \( +,-,*,\div \) since one can say that 1 has just the inverse -1 and that \( 1-\frac{2}{2} \neq 0\), but then one would be thinking about \( \mathbb Q \) as a weird alphabet. #grouptheory

I wrote a blog post on how to work out the injectivity radii of the unitary groups. If you like getting results in differential geometry and group theory with (mostly) just linear algebra, this one's for you! https://sethaxen.com/blog/2023/02/the-injectivity-radii-of-the-unitary-groups/

#DiffGeo #GroupTheory #LinearAlgebra

Anyone know anything about solving a 7x7x7 Rubik's cube?

I've got three centres down and I am looking for a hint (and just a hint!) if this is on the right track.

If I should be doing something else, I would appreciate a gentle nudge in another directions.

#RubiksCube #GroupTheory

Very good introduction to abstract algebra. Just one remark: why don't such guides cover categories?

https://www.youtube.com/watch?v=IP7nW_hKB7I&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6

#math #grouptheory #abstractalgebra

Relational PK-Nets for Transformational Music Analysis
by Alexandre Popoff, Moreno Andreatta, Andree Ehresmann
https://arxiv.org/abs/1611.02249
#categorytheory #musictheory #grouptheory

New maintenance release for GAP v4.12.2

https://www.gap-system.org/

#computer #algebra #grouptheory #semigroup

#introduction
I am a postdoc at the Princeton Neuroscience Institute, working with Tim Buschman and my brilliant lab mates on transformational geometry in the brain. We ask what are various classes of transformations brains utilize to compute for different tasks, in different contexts and with different objectives. Our main tools consist of linear algebra, differential geometry and group theory. 📐
#Neuroscience #WorkingMemory #GroupTheory #BrainGeometry #Geometry

There's actually not much on #GroupTheory used for the same, all I could find in a (brief) search was the defunct International Society for Group Theory in Cognitive Science

https://web.archive.org/web/20170618215304/http://www.rci.rutgers.edu/~mleyton/GT.htm

the only name i recognize on there are Michael Leyton, and Jean Petitot (who uses things like #ContactGeometry for neurogeometry). Eloise Carlton's work seems cool: https://www.sciencedirect.com/science/article/abs/pii/002224969090001P "Psychologically simple motions as geodesic paths I. Asymmetric objects"

Playing with a Rubik's Cube I noticed —

when starting at a certain state, doing a few twists and repeating these over and over again, you will allways come back to the state you started in.

I'm not sure whether this is interesting or trivial. Is this provable?

#Rubiks #Cube #GroupTheory #HowToSurviveBoringMeetings

#Introduction

Hi! I'm Null (they/them), a #nonbinary #trans #gamedev, #digifu #musician, and #math and #programming nerd. I am a bit of a bard (in the jack-of-all-trades sense, the musical sense, the cheery Wandersong bard personality sense, the preferred D&D class sense, among probably other senses). My current interests include #grouptheory, #functionalprogramming, #origami, #cartoons, #calligraphy, #sushi, #linguistics, #indiegames, #puzzlehunts, #accessibility, #musictheory.