# Group Theory Published articles for Group Theory. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## My Graduate Career in Math DevFeed: [My Graduate Career in Math]() Original publisher: [Read original article]() Published: 2025-05-12T18:35:57Z Content type: opinion Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Computer science](), [graph theory]() Tags: [computer-science](), [education](), [essay](), [essays](), [game-theory](), [graph-theory](), [group-theory](), [linear-algebra](), [math](), [mathematics](), [retrospective](), [topology](), [university]() ### AI overview An autobiographical essay about the author's transition from computer science to mathematics at Cal Poly, including university coursework, study abroad in Budapest, and reflections on the intellectual environment and an early group theory project. ### Source excerpt Editor's note: This essay was originally published on Medium on 2016-03-05. I have made minor edits in this republishing and added a few small retrospective notes. 2010-2011 (Year 0) I had just switched my major at Cal Poly State University from computer science to math. I wanted to double major but California was in a budget crisis and a few weeks before I tried submitting my double-major request the Provost for the CSU system put a blanket ban on double majors. ## Socks, a matching game based on an additive combinatorics problem DevFeed: [Socks, a matching game based on an additive combinatorics problem]() Original publisher: [Read original article]() Published: 2023-10-14T06:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math](), [structure]() Tags: [additive-combinatorics](), [board-games](), [card-game](), [combinatorics](), [game](), [games](), [group-theory](), [math](), [mathematics](), [structure](), [xor]() ### AI overview This article introduces Socks, a matching game modeled using six-dimensional binary vectors and additive combinatorics. It proves that any seven cards contain a valid zero-summing set, while six specific cards can avoid one, so the minimum guarantee is seven cards. ### Source excerpt Can you find a set of cards among these six, such that the socks on the chosen cards can be grouped into matching pairs? (Duplicate pairs of the same sock are OK) Spoilers: If the cards are indexed as 1 2 3 4 5 6 Then the following three subsets work: $\{ 1, 2, 4, 5, 6 \}$, $\{ 2, 3, 6 \}$, and $\{ 1, 3, 4, 5 \}$. ## The Gadget Decomposition in FHE DevFeed: [The Gadget Decomposition in FHE]() Original publisher: [Read original article]() Published: 2021-12-11T13:57:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [FHE](), [homomorphic encryption](), [Cryptography](), [Computing](), [data](), [Code](), [GitHub]() Tags: [bootstrapping](), [core](), [encryption](), [fhe](), [gadget-decomposition](), [group-theory](), [homomorphic-encryption](), [learning-with-errors](), [linear-algebra](), [lwe](), [mathematics](), [matrix](), [number-theory](), [operations](), [programming](), [python]() ### AI overview A tutorial on gadget decomposition in fully homomorphic encryption (FHE). It explains how GSW and related schemes use random noise, how homomorphic operations increase that noise, why bootstrapping is needed, and how gadget decomposition helps limit noise growth. ### Source excerpt Lately I've been studying Fully Homomorphic Encryption, which is the miraculous ability to perform arbitrary computations on encrypted data without learning any information about the underlying message. It's the most comprehensive private computing solution that can exist (and it does exist!). The first FHE scheme by Craig Gentry was based on ideal lattices and was considered very complex (I never took the time to learn how it worked). Some later schemes (GSW = Gentry-Sahai-Waters) are based on matrix multiplication, and are conceptually much simpler. ## Group Actions and Hashing Unordered Multisets DevFeed: [Group Actions and Hashing Unordered Multisets]() Original publisher: [Read original article]() Published: 2021-10-14T08:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [hashing](), [Programming]() Tags: [abelian-groups](), [group-actions](), [group-theory](), [groups](), [hashing](), [mathematics](), [multiset](), [multisets](), [practical](), [programming](), [xor]() ### AI overview The article introduces a result by Kevin Ventullo that applies group actions to hash functions for unordered sets and multisets. It explains why incremental, order-independent hashing is useful and describes collision-related weaknesses of addition and XOR approaches. ### Source excerpt I learned of a neat result due to Kevin Ventullo that uses group actions to study the structure of hash functions for unordered sets and multisets. This piqued my interest because a while back a colleague asked me if I could think of any applications of "pure" group theory to practical computer programming that were not cryptographic in nature. He meant, not including rings, fields, or vector spaces whose definitions happen to be groups when you forget the extra structure. ## A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details DevFeed: [A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details]() Original publisher: [Read original article]() Published: 2015-11-12T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [Algorithms, Complexity](), [graph theory](), [Computer science](), [math]() Tags: [algorithm](), [automorphism-groups](), [babai](), [breakthrough](), [complexity](), [computer-science](), [divide-and-conquer](), [graph-isomorphism](), [group-theory](), [johnson-graph](), [mathematics](), [permutation-groups](), [quasipolynomial](), [research]() ### AI overview This research-level blog post explains Laszlo Babai's claimed quasipolynomial-time algorithm for the Graph Isomorphism problem. Later updates report that an error was found, changing the result to sub-exponential time, followed by a claimed workaround restoring the quasipolynomial-time claim. ### Source excerpt Update 2017-01-09: Laci claims to have found a workaround to the previously posted error, and the claim is again quasipolynoimal time! Updated arXiv paper to follow. Update 2017-01-04: Laci has posted an update on his paper. The short version is that one small step of his analysis was not quite correct, and the result is that his algorithm is sub-exponential, but not quasipolynomial time. The fact that this took over a year to sort out is a testament to the difficulty of the mathematics and the skill of the mathematicians involved. ## The Fundamental Theorem of Algebra (with Galois Theory) DevFeed: [The Fundamental Theorem of Algebra (with Galois Theory)]() Original publisher: [Read original article]() Published: 2012-02-02T22:42:11Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [field](), [fundamental-theorem-of-algebra](), [galois-theory](), [group-theory](), [mathematics](), [polynomials](), [roots]() ### AI overview This mathematics post presents a Galois-theoretic proof strategy for the fundamental theorem of algebra. It assumes familiarity with field extensions, Galois theory, and group theory, and develops the argument using splitting fields, an intermediate extension of degree 2, Sylow subgroups, and the Galois correspondence. ### Source excerpt This post assumes familiarity with some basic concepts in abstract algebra, specifically the terminology of field extensions, and the classical results in Galois theory and group theory. The fundamental theorem of algebra has quite a few number of proofs (enough to fill a book!). In fact, it seems a new tool in mathematics can prove its worth by being able to prove the fundamental theorem in a different way. This series of proofs of the fundamental theorem also highlights how in mathematics there are many many ways to prove a single theorem, and in re-proving an established theorem we introduce new concepts and strategies. ## The Smallest Non-Cyclic Simple Group has Order 60 DevFeed: [The Smallest Non-Cyclic Simple Group has Order 60]() Original publisher: [Read original article]() Published: 2011-10-08T20:53:48Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [classification](), [group-theory](), [mathematics](), [primes](), [simple-groups](), [theory]() ### AI overview A mathematical proof examines simple groups of order less than 60 and establishes that the smallest non-cyclic simple group has order 60. It uses group theory concepts including normal subgroups, group actions, and the Sylow theorems. ### Source excerpt Preamble: This proof is not particularly elegant or insightful. However, it belongs in this gallery for two reasons. First, it is an example of the goal of most mathematics: to classify things. In the same way that all natural numbers can be built up from primes, every group can be built up from simple groups. So if we want to understand all groups, it suffices to understand the simple ones. Indeed, this project has been the collective goal of hundreds of mathematicians for the past hundred years, culminating in one awesomely gargantuan theorem: The Classification of Finite Simple Groups.