Upgoer 5: The main topic of my thesis using the ten thousand most commonly used words

[alias_sqbr]

I once wrote a book about how to work out the name of a thing called a "black box group."

A "group" is a group of things you can add together so that when you add any two things in the group you get another thing in the group. Every group has a "nothing" that does nothing when you add it, and everything in a group has it's own "mirror" that gives you nothing when you add it.

Numbers are a kind of group, since if you add two numbers you always get a number and not, say, a house. The "nothing" of numbers is...nothing! Six added to nothing is six and so on. The "mirror" of six is the number "nothing take away six", since six added to nothing take away six is nothing.

Another group is "turn this thing the other way up, or leave it alone": If you turn something the other way up two times, it's the same as leaving it alone (try it out and see!), so no matter how many times you add those two things together you stay inside the group. See if you can work out what the nothing and mirrors are!

A "black box" group is a group you don't know much about. It's like all you have is a black box that tells you what you get if you add two things in the group together, but you can't look at the things in the group and work out just by looking to see if they are, say, the number six, or "turn this thing the other way up" and so on.

You can still work out what a group really is if all you have is this black box. Say I have a black box group with two things in it, and the black box tells me they are called dog and cat. The black box tells me that dog added to dog is dog, dog added to cat is cat, and cat added to cat is dog. I can look at that and see that it acts the same way as the group "turn this thing the other way up, or leave it alone" if I give "leave it alone" the name "dog" and "turn it the other way up" the name "cat".

My job was to take a black box group and see if it was really just the same as another group I already knew. It took a long time, the groups I was working with were very big!

(Inspired by kaz's charming description of all of Algebra. Here's the Up-Goer Five text editor if you want to have a go)

Comments

[quantumreality]

Hey! I am coming to this like OVER A DECADE LATER

I somehow stumbled across this from a tumblr link and I find the concept to be quite interesting (using only the most common English words to try and describe a thing). :)

I figured out you were discussing group theory and how certain operations reveal symmetries in a group (which can be thought of as leaving the object in question unchanged after performing such an operation).

What was the "black box" group you were working on, though? :)

[alias_sqbr]

Hello! Very cool to know my old post still gets read! I was working on a couple of different sporadic simple groups, it's been so long I couldn't remember which off the top of my head so actually cracked open my thesis for the first time in A While... ok says here Higman Sims, the Mathei groups M11, M12, and M22, and the Janko group J2.