John Carlos Baez on Nostr: Now I'll the same stuff like a mathematician: we saw that the dihedral group with 16 ...

Now I'll the same stuff like a mathematician: we saw that the dihedral group with 16 elements, also called 𝐷₁₆, has a presentation with generators 𝑟,𝑠 and relations

𝑟⁸ = 𝑠² = (𝑟𝑠)² = 𝐼,

We also saw its Cayley diagram, labeling each group element by its order.

Quicker, but fewer people will understand!

Now for something a bit less known. The group we just saw has an evil twin, another group with 16 elements, called the 'quasidihedral group'. Only one of the relations is different: now we have

𝑠𝑟𝑠 = 𝑟³

This makes the Cayley diagram look like an 8-pointed star inside an octagon!

I heard about this group from Ianagol (npub1jgy…rkt6), and I instantly looked it up on Wikipedia, where they have these nice pictures:

https://en.wikipedia.org/wiki/Quasidihedral_group

In fact ost finite groups have a size that's a power of two! So there are a *lot* of different groups with 16 elements - namely, 14 of them. So, if you were stuck on a desert island, you could have fun figuring out what they all are, and drawing the Cayley diagrams of all 14. In fact if the world keeps going to hell, I might go to a desert island and do just that.

(2/2)