Browsed by
Category: Group Theory

Cayley’s Theorem: Powerful Permutations

Cayley’s Theorem: Powerful Permutations

We've discussed before how powerful isomorphisms can be, when we find them. Finding isomorphisms "from scratch" can be quite a challenge. Thankfully, Arthur Cayley proved one of the classic theorems of modern algebra that can help make our lives a bit easier. We'll explore this theorem and note the power of groups of permutations.  (more…)

Isomorphisms: Making Mathematics More Convenient

Isomorphisms: Making Mathematics More Convenient

Much of pure mathematics exists to simplify our world, even if it means entering an abstract realm (or creating one) to do it. The isomorphism is one of the most powerful tools for discovering structural similarities (or that two groups are identical structurally) between two groups that on the surface look completely unrelated. In this post,…

All the Same Opposites

All the Same Opposites

Editor's note: see this appendix for supporting proofs. Fields are among the most convenient algebraic structures, preserving much of the arithmetic we know and love from familiar fields like the rationals and the real numbers . Now, it is unnecessary that a set possess infinitely many elements to possibly constitute a field (under the right…

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

The introduction to coding theory in this post will now allow us to explore some more interesting topics in coding theory, courtesy of Pinter's A Book of Abstract Algebra. We'll introduce the notion of a code, informations, and parity check equations. Most communication channels are noisy to some extent, which means that a transmitted codeword may have…

Group Theory, XOR, and Binary Codes: Introducing Coding Theory

Group Theory, XOR, and Binary Codes: Introducing Coding Theory

Binary codes and bit-wise operations are fundamental in computer science. Whatever device you are running today works because of binary codes, and the bit-wise operations AND, OR, NOT, and XOR. (A fun exercise, prove to yourself that each of these rules meets the definition of an operation. If you need a refresher, check out this…