JRG's Blog

~😐 = 🫥

~🫥 = 😐

😐ʌ😐 = 😐

😐ʌ🫥 = 🫥

🫥ʌ😐 = 🫥

🫥ʌ🫥 = 🫥

😐v😐 = 😐

😐v🫥 = 😐

🫥v😐 = 😐

"reflective algebra is an incredibly powerful branch of mathematics that extends the usual operations on numbers to geometric mirrors.

in school, we first learned elementary arithmetic, where we add, subtract, multiply and divide numbers. we then later learned about elementary algebra, where we start describing these operations on arbitrary numbers, this leads to the finding of equations that are valid for all numbers:

a+b = b+a

a(b+c) = ab+ac

similarly, there is reflective arithmetic, where we add, subtract, multiply and divide mirrors. this leads to reflective algebra, where we start describing these operations on arbitrary mirrors, this leads to the finding of equations that are valid for all mirrors:

x̂x̂ = 1

xy = x•y + xʌy

many problems in geometry, computer graphics, etc, are reflective problems, so reflective algebra provides solutions in all of these fields.

so then, if reflective algebra is so useful, then how come it doesn't seem to get used that much? one of the reasons is a lack of introductary material for it. most introductions to reflective algebra are only found in esoteric corners of the online math community. this causes math teachers to not care about reflective algebra, and thus there is no one to teach the students.

so the handful who know about this obscure branch of math had to learn how to solve reflective problems without reflective algebra before learning to solve them with reflective algebra. and as any member of the former would tell you from experience using the formalism, reflective algebra doesn't get more complicated that what would would see using something else, and in many cases, it is simpler.

🦊alright, it’s time for the groups of order 6

🐱well for starters there’s C6, a.k.a.

🐱🐱🐱🐱🐱🐱=🆔

🦊yes. but there’s another group of order 6, but before i get to that, let’s poke around with C6 for a bit. 6 factors as 2x3, so we can see what happens if we multiply C2xC3

🐱so that’s, uh…

🐱🐱🐱=🆔

🦊🦊=🆔

🐱🦊=🦊🐱

🐱is that the second group of order 6?

🦊it’s not, and the reason why is because of the fundamental theorem of finite abelian groups:

every finite abelian group is the unique product of prime power cycles

🐱prime powers… uh, like how C2xC2 is not isomorphic to C4?

🦊yes. cycles can combine into bigger cycles only if their orders are coprime, so C2xC3 is isomorphic to C6, but C2xC2 is not isomorphic to C4

🐱but why is that the case?

🦊technically its due to things like homomorphisms and the orders of the elements, but i’ll leave that as an exercise for the readers

🐱whatever. so now can we get to what the order group of order 6 is?

🦊it has a property that we haven’t seen before. all the groups we’ve seen thus far have been abelian, meaning that 🐱🦊=🦊🐱 for any two elements 🐱 and 🦊. the other group of order 6 is the smallest non abelian group, so 🐱🦊≠🦊🐱 for certain values of 🐱 and 🦊

🐱how is it constructed tho?

🦊hey, remember back when we went over automorphisms? we looked at C3 and a map that took elements of C3 to their inverses:

🐱🐱🐱=🆔

🦊🦊=🆔

🦊(🐱)=🐱🐱

🐱ah yes, i remember that now that you’ve mentioned it

🦊the non abelian group of order 6 is derived from a set of equations similar to those, but the bottom equation is modded so that 🦊 is an element of the group itself:

🐱🐱🐱=🆔

🦊🦊=🆔

🐱🦊=🦊🐱🐱

🐱so this is like C2xC3, but since the C2 element twists the C3 cycle, for any two distinct arbitrary elements of that group 🐰 and 🐶, 🐰🐶 doesn’t always equal 🐶🐰

🦊yes, exactly! also, that group has its own shorthand name, it’s called D3, as in the dihedral group of a triangle. also, for groups that are 2 times a prime p, there’s just C2p and Dp. just as cycle groups are of the general form:

n🐱=🆔

🦊dihedral groups are generally of the form:

n🐱=🆔

🦊🦊=🆔

🐱🦊=🦊~🐱

🐱wait, 2 times a prime p… 2 is itself a prime, so for p=2, there’s C4:

🐱🐱🐱🐱=🆔

🐱and D2, which is isomorphic to C2xC2:

🐱🐱=🆔

🦊🦊=🆔

🐱🦊=🦊🐱

🦊anyway, let’s move on. for order 7 there’s just C7, i.e.

7🐱=🆔

🦊now to the group of order 5, as there’s just C5, i.e.

🐱🐱🐱🐱🐱=🆔

🐱why is that the only one?

🦊good question. it’s to do with subgroups and the orders of a group’s elements. all of the groups we’ve seen thus far except for C2xC2 are cyclic, meaning all their elements are powers of a single generator:

C1: 🐱=🆔

C2: 🐱🐱=🆔

C3: 🐱🐱🐱=🆔

C4: 🐱🐱🐱🐱=🆔

C5: 🐱🐱🐱🐱🐱=🆔

C2xC2:

🐱🐱=🆔

🦊🦊=🆔

🐱🦊=🦊🐱

🐱wait, C2xC2 has the powers of 🐱 generate C2, which is a group of order 2, which is a factor of 4, but that’s obvious gives that C2xC2 is literally just a product of smaller groups?

🦊of course that sounds obvious, but you’re getting at the idea behind what subgroups and the orders elements within a group are. the order of an element is the number of elements reached by the powers of that element, and the order of the resulting subgroup always is a number that divides the order of the original group.

🐱now can we get to why C5 is the group of order 5? …wait, every number of elements we’ve looked at thus far have had one group each except for order 4

🦊yes, and this is where the why fits in. so now i’ll explain why prime orders have only cyclic groups. suppose there’s the identity 🆔 and an element a group of order p such that 🐱≠🆔. the order of 🐱 thus must be either 1 or p. it can’t be 1, since that’d make 🐱=🆔 (the definition of C1, which happens to be the only group of order 1, but regardless), so the order of 🐱 must be p

🐱so there’s only one group for the orders 1 and the prime numbers 2, 3, 5, 7, 11, etc, and they’re all cyclic?

🦊yes, exactly. now it’s at this point i’d start going over the groups of order 6, but tbh i feel like i need to take a nap, so you’re free to go for now

🐱woohoo! class is dismissed early! *walks out*

🐱*wakes up* …last thing i remember was something about C2xC2 and isomorphisms

🦊C2xC2 and C4 not being isomorphic, yes

🐱because there’s no H() between them such that that H() has an inverse?

🦊well, technically yes, but to concretely show why C2xC2 and C4 are not isomorphic, we can look at the inverses:

in C4, there’s ~🐱=🐱🐱🐱 and ~(🐱🐱)=🐱🐱, but in C2xC2 there’s ~🐱=🐱, ~🦊=🦊 and ~(🐱🦊)=🐱🦊

🐱so C4 has only one self-inverse element while C2xC2 has three