4 ms·

Groups: A Primer (2012)

39 points by subnaught 12y ago

mrcactu5 12y ago
great article i still think it's tough to find specific examples where symmetries help is in programming. semigroups represent the composition of code, and groups are just semigroups with an "undo" operation. e.g http://mmisamore.github.io/2014/09/17/Monogenic-Semigroups/ http://mmisamore.github.io/2014/09/17/Monogenic-Semigroups/ has anyone looked on github and drawn the diagrams to show their library is a group or semigroup?
j2kun 12y ago
As an example of a useful group in programming: elliptic curves used in cryptography.
- ky3 12y ago
  Z/Zn groups are a better example, whether seen in RSA or Diffie-Hellman. Elliptic curves require graduate-school math to properly appreciate. Being mathematically mysterious, they are still the subject of active research. Vanilla modular arithmetic is at least as old as Fermat, about 400 years. A breakthrough in ECC is big news. A breakthrough in the integers would flip the world upside down.
  - j2kun 12y ago
    I think you mean the multiplicative group of units of Z/nZ, because additively Z/nZ has much more structure than a group. Still, for my example and in the article, one should be motivated to study groups because they show up in places that you don't expect to find algebraic structure. Z/nZ and its variants are the obvious places to look. Elliptic curves are surprising. Also, I wrote a series on elliptic curves which doesn't need any graduate level mathematics to appreciate [1]. Of course if you want to make breakthroughs on elliptic curves (or any serious mathematical topic) you need graduate study. [1]: http://jeremykun.com/2014/02/08/introducing-elliptic-curves/ http://jeremykun.com/2014/02/08/introducing-elliptic-curves/
mpancia 12y ago
This is pretty good, although I am always a bit worried that it is difficult for people to internalize this stuff without a real drive to get into abstract algebra. As a side note, I think a really nice way to go (dovetailing off of the ending group actions section) is to use group actions and this lemma (https://qchu.wordpress.com/2013/07/09/the-p-group-fixed-point-theorem/ https://qchu.wordpress.com/2013/07/09/the-p-group-fixed-poin...) to prove a bunch of cool stuff. It really unifies all of the basic stuff and makes it clear that finite group theory is really entirely about symmetries/actions.

39 points by subnaught 12y ago

great article i still think it's tough to find specific examples where symmetries help is in programming. semigroups represent the composition of code, and groups are just semigroups with an "undo" operation. e.g http://mmisamore.github.io/2014/09/17/Monogenic-Semigroups/ http://mmisamore.github.io/2014/09/17/Monogenic-Semigroups/ has anyone looked on github and drawn the diagrams to show their library is a group or semigroup?

j2kun 12y ago

As an example of a useful group in programming: elliptic curves used in cryptography.

ky3 12y ago
Z/Zn groups are a better example, whether seen in RSA or Diffie-Hellman. Elliptic curves require graduate-school math to properly appreciate. Being mathematically mysterious, they are still the subject of active research. Vanilla modular arithmetic is at least as old as Fermat, about 400 years. A breakthrough in ECC is big news. A breakthrough in the integers would flip the world upside down.
- j2kun 12y ago
  I think you mean the multiplicative group of units of Z/nZ, because additively Z/nZ has much more structure than a group. Still, for my example and in the article, one should be motivated to study groups because they show up in places that you don't expect to find algebraic structure. Z/nZ and its variants are the obvious places to look. Elliptic curves are surprising. Also, I wrote a series on elliptic curves which doesn't need any graduate level mathematics to appreciate [1]. Of course if you want to make breakthroughs on elliptic curves (or any serious mathematical topic) you need graduate study. [1]: http://jeremykun.com/2014/02/08/introducing-elliptic-curves/ http://jeremykun.com/2014/02/08/introducing-elliptic-curves/

mpancia 12y ago

This is pretty good, although I am always a bit worried that it is difficult for people to internalize this stuff without a real drive to get into abstract algebra. As a side note, I think a really nice way to go (dovetailing off of the ending group actions section) is to use group actions and this lemma (https://qchu.wordpress.com/2013/07/09/the-p-group-fixed-point-theorem/ https://qchu.wordpress.com/2013/07/09/the-p-group-fixed-poin...) to prove a bunch of cool stuff. It really unifies all of the basic stuff and makes it clear that finite group theory is really entirely about symmetries/actions.