- #1
- 19,904
- 26,029
Last edited:
The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement that the order of two consecutive operations does not matter: ##(1+2)+3=1+(2+3).## That's it. The concept of a group is so simple that I still wonder why it wasn't part of my syllabus at school. And, yet, it covers such different sets like the integers, the hours that the big hand counts, the symmetries in a crystal, the Caesar cipher, or a light switch which is the basis of our electronic world. However, few requirements allow many additional, more specific refinements. In the case of groups, we arrive at strange-sounding results like the fact that the largest finite, and simple, sporadic group has
$$
808017424794512875886459904961710757005754368000000000
$$
many elements. This article is meant to shed some light on the betweens of a light switch and a group with more than ##8\cdot 10^{53}## elements that mathematicians dare to call simple. At least, they also call it the monster group, and the second largest finite, simple, sporadic group with its
$$
4154781481226426191177580544000000
$$
many elements baby monster group. And to be honest, even the simple fact that they found them is still a mystery to me.
This article explains fundamental concepts and only lists the mysterious results. It is meant as an introduction to group theory rather than a treatment of the many special areas into which group theory has branched out. Many statements especially in the sections about examples and structures can be verified by the readers if they wish to practice typical conclusions in group theory.
I meant that ERH is the interesting conjecture for cryptography and number theory, RH without ERH not so much.martinbn said:What do you mean when you say that the extended Riemann hypothesis is believed to be proven?